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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Finite element method</span></span>
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</style><div role="note" class="hatnote navigation-not-searchable">"Finite element" redirects here. For the elements of a <a href="Poset" class="mw-redirect" title="Poset">poset</a>, see <a href="Compact_element" title="Compact element">compact element</a>.</div>

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title" style="background:#ccccff;display:block;margin-bottom:0.2em;"><a href="Differential_equation" title="Differential equation">Differential equations</a></th></tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;">
Scope</th></tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Fields</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="padding-bottom:0;">
<div class="hlist"><ul><li><a href="Natural_science" title="Natural science">Natural sciences</a></li><li><a href="Engineering" title="Engineering">Engineering</a></li></ul></div></th></tr><tr><td class="sidebar-content hlist" style="padding-bottom:0.6em;">
<ul><li><a href="Astronomy" title="Astronomy">Astronomy</a></li>
<li><a href="Physics" title="Physics">Physics</a></li>
<li><a href="Chemistry" title="Chemistry">Chemistry</a></li>
<li><br><a href="Biology" title="Biology">Biology</a></li>
<li><a href="Geology" title="Geology">Geology</a></li></ul></td>
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<a href="Applied_mathematics" title="Applied mathematics">Applied mathematics</a></th></tr><tr><td class="sidebar-content hlist" style="padding-bottom:0.6em;">
<ul><li><a href="Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></li>
<li><a href="Chaos_theory" title="Chaos theory">Chaos theory</a></li>
<li><a href="Dynamical_systems" class="mw-redirect" title="Dynamical systems">Dynamical systems</a></li></ul></td>
</tr><tr><th class="sidebar-heading" style="padding-bottom:0;">
<a href="Social_science" title="Social science">Social sciences</a></th></tr><tr><td class="sidebar-content hlist" style="padding-bottom:0.6em;;padding-bottom:0;">
<ul><li><a href="Economics" title="Economics">Economics</a></li>
<li><a href="Population_dynamics" title="Population dynamics">Population dynamics</a></li></ul></td>
</tr></tbody></table>
<hr>
<a href="List_of_named_differential_equations" title="List of named differential equations">List of named differential equations</a></div></div></td>
</tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;;display:block;margin-top:0.1em;">
Classification</th></tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Types</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content">
<div class="hlist">
<ul><li><a href="Ordinary_differential_equation" title="Ordinary differential equation">Ordinary</a></li>
<li><a href="Partial_differential_equation" title="Partial differential equation">Partial</a></li>
<li><a href="Differential-algebraic_system_of_equations" title="Differential-algebraic system of equations">Differential-algebraic</a></li>
<li><a href="Integro-differential_equation" title="Integro-differential equation">Integro-differential</a></li>
<li><a href="Fractional_differential_equations" class="mw-redirect" title="Fractional differential equations">Fractional</a></li>
<li><a href="Linear_differential_equation" title="Linear differential equation">Linear</a></li>
<li><a href="Non-linear_differential_equation" class="mw-redirect" title="Non-linear differential equation">Non-linear</a></li></ul>
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</tr><tr><th class="sidebar-heading">
By variable type</th></tr><tr><td class="sidebar-content">
<ul><li><a href="Dependent_and_independent_variables" title="Dependent and independent variables">Dependent and independent variables</a></li></ul>
<div class="hlist">
<ul><li><a href="Autonomous_differential_equation" class="mw-redirect" title="Autonomous differential equation">Autonomous</a></li>
<li>Coupled&nbsp;/ Decoupled</li>
<li><a href="Exact_differential_equation" title="Exact differential equation">Exact</a></li>
<li><a href="Homogeneous_differential_equation" title="Homogeneous differential equation">Homogeneous</a>&nbsp;/ <a href="Non-homogeneous_differential_equation" class="mw-redirect" title="Non-homogeneous differential equation">Nonhomogeneous</a></li></ul>
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Features</th></tr><tr><td class="sidebar-content">
<div class="hlist">
<ul><li><a href="Ordinary_differential_equation#Definitions" title="Ordinary differential equation">Order</a></li>
<li><a href="Differential_operator" title="Differential operator">Operator</a></li></ul>
</div>
<ul><li><a href="Notation_for_differentiation" title="Notation for differentiation">Notation</a></li></ul></td>
</tr></tbody></table></div></div></td>
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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Relation to processes</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;">
<ul><li><a href="Difference_equation" class="mw-redirect" title="Difference equation">Difference <span style="font-size: 85%;">(discrete analogue)</span></a></li></ul>
<div class="hlist">
<ul><li><a href="Stochastic_differential_equation" title="Stochastic differential equation">Stochastic</a>
<ul><li><a href="Stochastic_partial_differential_equation" title="Stochastic partial differential equation">Stochastic partial</a></li></ul></li>
<li><a href="Delay_differential_equation" title="Delay differential equation">Delay</a></li></ul>
</div></div></div></td>
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Solution</th></tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Existence and uniqueness</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;">
<ul><li><a href="Picard%E2%80%93Lindel%C3%B6f_theorem" title="Picard–Lindelöf theorem">Picard–Lindelöf theorem </a></li>
<li><a href="Peano_existence_theorem" title="Peano existence theorem">Peano existence theorem</a></li>
<li><a href="Carath%C3%A9odory's_existence_theorem" title="Carathéodory's existence theorem">Carathéodory's existence theorem</a></li>
<li><a href="Cauchy%E2%80%93Kowalevski_theorem" class="mw-redirect" title="Cauchy–Kowalevski theorem">Cauchy–Kowalevski theorem</a></li></ul></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">General topics</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><div class="hlist">
<ul><li><a href="Initial_condition" title="Initial condition">Initial conditions</a></li>
<li><a href="Boundary_value_problem" title="Boundary value problem">Boundary values</a>
<ul><li><a href="Dirichlet_boundary_condition" title="Dirichlet boundary condition">Dirichlet</a></li>
<li><a href="Neumann_boundary_condition" title="Neumann boundary condition">Neumann</a></li>
<li><a href="Robin_boundary_condition" title="Robin boundary condition">Robin</a></li>
<li><a href="Cauchy_problem" title="Cauchy problem">Cauchy problem</a></li></ul></li>
<li><a href="Wronskian" title="Wronskian">Wronskian</a></li>
<li><a href="Phase_portrait" title="Phase portrait">Phase portrait</a></li>
<li><a href="Lyapunov_stability" title="Lyapunov stability">Lyapunov</a>&nbsp;/ <a href="Asymptotic_stability" class="mw-redirect" title="Asymptotic stability">Asymptotic</a>&nbsp;/ <a href="Exponential_stability" title="Exponential stability">Exponential stability</a></li>
<li><a href="Rate_of_convergence" title="Rate of convergence">Rate of convergence</a></li>
<li><span class="nowrap"><a href="Power_series_solution_of_differential_equations" title="Power series solution of differential equations">Series</a>&nbsp;/ Integral solutions</span></li>
<li><a href="Numerical_integration" title="Numerical integration">Numerical integration</a></li>
<li><a href="Dirac_delta_function" title="Dirac delta function">Dirac delta function</a></li></ul>
</div></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Solution methods</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><div class="hlist">
<ul><li>Inspection</li>
<li><a href="Method_of_characteristics" title="Method of characteristics">Method of characteristics</a></li>
<li><br><a href="Euler_method" title="Euler method">Euler</a></li>
<li><a href="Exponential_response_formula" title="Exponential response formula">Exponential response formula</a></li>
<li><a href="Finite_difference_method" title="Finite difference method">Finite difference</a>&nbsp;<span style="font-size: 85%;">(<a href="Crank%E2%80%93Nicolson_method" title="Crank–Nicolson method">Crank–Nicolson</a>)</span></li>
<li>
<ul><li><a href="Infinite_element_method" title="Infinite element method">Infinite element</a></li></ul></li>
<li><a href="Finite_volume_method" title="Finite volume method">Finite volume</a></li>
<li><a href="Galerkin_method" title="Galerkin method">Galerkin</a>
<ul><li><a href="Petrov%E2%80%93Galerkin_method" title="Petrov–Galerkin method">Petrov–Galerkin</a></li></ul></li>
<li><a href="Green's_function" title="Green's function">Green's function</a></li>
<li><a href="Integrating_factor" title="Integrating factor">Integrating factor</a></li>
<li><a href="Integral_transform" title="Integral transform">Integral transforms</a></li>
<li><a href="Perturbation_theory" title="Perturbation theory">Perturbation theory</a></li>
<li><a href="Runge%E2%80%93Kutta_methods" title="Runge–Kutta methods">Runge–Kutta</a></li></ul>
</div>
<ul><li><a href="Separation_of_variables" title="Separation of variables">Separation of variables</a></li>
<li><a href="Method_of_undetermined_coefficients" title="Method of undetermined coefficients">Undetermined coefficients</a></li>
<li><a href="Variation_of_parameters" title="Variation of parameters">Variation of parameters</a></li></ul></div></div></td>
</tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;">
People</th></tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">List</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><div class="hlist" style="padding-top:0.5em">
<ul><li><a href="Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li>
<li><a href="Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Gottfried Leibniz</a></li>
<li><a href="Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a></li>
<li><a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li>
<li><a href="Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a></li>
<li><a href="J%C3%B3zef_Maria_Hoene-Wro%C5%84ski" title="Józef Maria Hoene-Wroński">Józef Maria Hoene-Wroński</a></li>
<li><a href="Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a></li>
<li><a href="Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li>
<li><a href="George_Green_(mathematician)" title="George Green (mathematician)">George Green</a></li>
<li><a href="Carl_David_Tolm%C3%A9_Runge" class="mw-redirect" title="Carl David Tolmé Runge">Carl David Tolmé Runge</a></li>
<li><a href="Martin_Kutta" title="Martin Kutta">Martin Kutta</a></li>
<li><a href="Rudolf_Lipschitz" title="Rudolf Lipschitz">Rudolf Lipschitz</a></li>
<li><a href="Ernst_Lindel%C3%B6f" class="mw-redirect" title="Ernst Lindelöf">Ernst Lindelöf</a></li>
<li><a href="%C3%89mile_Picard" title="Émile Picard">Émile Picard</a></li>
<li><a href="Phyllis_Nicolson" title="Phyllis Nicolson">Phyllis Nicolson</a></li>
<li><a href="John_Crank" title="John Crank">John Crank</a></li></ul>
</div></div></div></td>
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<p><b>Finite element method</b> (<b>FEM</b>) is a popular method for numerically solving <a href="Differential_equation" title="Differential equation">differential equations</a> arising in engineering and <a href="Mathematical_models" class="mw-redirect" title="Mathematical models">mathematical modeling</a>. Typical problem areas of interest include the traditional fields of <a href="Structural_analysis" title="Structural analysis">structural analysis</a>, <a href="Heat_transfer" title="Heat transfer">heat transfer</a>, <a href="Fluid_flow" class="mw-redirect" title="Fluid flow">fluid flow</a>, mass transport, and <a href="Electromagnetic_potential" class="mw-redirect" title="Electromagnetic potential">electromagnetic potential</a>. Computers are usually used to perform the calculations required. With high-speed <a href="Supercomputer" title="Supercomputer">supercomputers</a>, better solutions can be achieved and are often required to solve the largest and most complex problems.
</p><p>FEM is a general <a href="Numerical_analysis" title="Numerical analysis">numerical method</a> for solving <a href="Partial_differential_equations" class="mw-redirect" title="Partial differential equations">partial differential equations</a> in two- or three-space variables (i.e., some <a href="Boundary_value_problem" title="Boundary value problem">boundary value problems</a>). There are also studies about using FEM to solve high-dimensional problems.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> To solve a problem, FEM subdivides a large system into smaller, simpler parts called <b>finite elements</b>. This is achieved by a particular space <a href="Discretization" title="Discretization">discretization</a> in the space dimensions, which is implemented by the construction of a <a href="Types_of_mesh" title="Types of mesh">mesh</a> of the object: the numerical domain for the solution that has a finite number of points. FEM formulation of a boundary value problem finally results in a system of <a href="Algebraic_equation" title="Algebraic equation">algebraic equations</a>. The method approximates the unknown function over the domain.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. FEM then approximates a solution by minimizing an associated error function via the <a href="Calculus_of_variations" title="Calculus of variations">calculus of variations</a>.
</p><p>Studying or <a href="Analysis" title="Analysis">analyzing</a> a phenomenon with FEM is often referred to as <b>finite element analysis</b> (FEA).
</p>
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<div class="mw-heading mw-heading2"><h2 id="Basic_concepts">Basic concepts</h2></div>
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</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:608px;max-width:608px"><div class="trow"><div class="tsingle" style="width:302px;max-width:302px"><div class="thumbimage"><span typeof="mw:File"></span></div><div class="thumbcaption">FEM <a href="Polygon_mesh" title="Polygon mesh">mesh</a> created by an analyst before finding a solution to a <a href="Magnetism" title="Magnetism">magnetic</a> problem using FEM software. Colors indicate that the analyst has set material properties for each zone, in this case, a <a href="Electrical_conductor" title="Electrical conductor">conducting</a> wire coil in orange; a <a href="Ferromagnetism" title="Ferromagnetism">ferromagnetic</a> component (perhaps <a href="Iron" title="Iron">iron</a>) in light blue; and air in grey. Although the geometry may seem simple, it would be very challenging to calculate the magnetic field for this setup without FEM software using <a href="Closed-form_expression" title="Closed-form expression">equations alone</a>.</div></div><div class="tsingle" style="width:302px;max-width:302px"><div class="thumbimage"><span typeof="mw:File"></span></div><div class="thumbcaption">FEM solution to the problem at left, involving a <a href="Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">cylindrically</a> shaped <a href="Magnetic_shielding" class="mw-redirect" title="Magnetic shielding">magnetic shield</a>. The <a href="Ferromagnetism" title="Ferromagnetism">ferromagnetic</a> cylindrical part shields the area inside the cylinder by diverting the magnetic field <a href="Electromagnet" title="Electromagnet">created</a> by the coil (rectangular area on the right). The color represents the <a href="Norm_(mathematics)" title="Norm (mathematics)">amplitude</a> of the <a href="Magnetic_field#Definitions,_units,_and_measurement" title="Magnetic field">magnetic flux density</a>, as indicated by the scale in the inset legend, red being high amplitude. The area inside the cylinder is low amplitude (dark blue, with widely spaced lines of magnetic flux), which suggests that the shield is performing as it was designed to.</div></div></div></div></div>
<p>The subdivision of a whole domain into simpler parts has several advantages:<sup id="cite_ref-:0_3-0" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>
</p>
<ul><li>Accurate representation of complex geometry;</li>
<li>Inclusion of dissimilar material properties;</li>
<li>Easy representation of the total solution; and</li>
<li>Capture of local effects.</li></ul>
<p>A typical approach using the method involves the following steps:
</p>
<ol><li>Dividing the domain of the problem into a collection of subdomains, with each subdomain represented by a set of element equations for the original problem.</li>
<li>Systematically recombining all sets of element equations into a global system of equations for the final calculation.</li></ol>
<p>The global system of equations uses known solution techniques and can be calculated from the <a href="Initial_value" class="mw-redirect" title="Initial value">initial values</a> of the original problem to obtain a numerical answer.
</p><p>In the first step above, the element equations are simple equations that locally approximate the original complex equations to be studied, where the original equations are often <a href="Partial_differential_equation" title="Partial differential equation">partial differential equations</a> (PDEs). To explain the approximation of this process, FEM is commonly introduced as a special case of the <a href="Galerkin_method" title="Galerkin method">Galerkin method</a>. The process, in mathematical language, is to construct an integral of the <a href="Inner_product" class="mw-redirect" title="Inner product">inner product</a> of the residual and the <a href="Weight_function" title="Weight function">weight functions</a>; then, set the integral to zero. In simple terms, it is a procedure that minimizes the approximation error by fitting trial functions into the PDE. The residual is the error caused by the trial functions, and the weight functions are <a href="Polynomial" title="Polynomial">polynomial</a> approximation functions that project the residual. The process eliminates all the spatial derivatives from the PDE, thus approximating the PDE locally using the following:
</p>
<ul><li>a set of <a href="Algebraic_equations" class="mw-redirect" title="Algebraic equations">algebraic equations</a> for <a href="Steady_state" title="Steady state">steady-state</a> problems; and</li>
<li>a set of <a href="Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equations</a> for <a href="Transient_state" title="Transient state">transient</a> problems.</li></ul>
<p>These equation sets are element equations. They are <a href="Linear" class="mw-redirect" title="Linear">linear</a> if the underlying PDE is linear and vice versa. Algebraic equation sets that arise in the steady-state problems are solved using <a href="Numerical_linear_algebra" title="Numerical linear algebra">numerical linear algebraic</a> methods. In contrast, <a href="Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equation</a> sets that occur in the transient problems are solved by numerical integrations using standard techniques such as <a href="Euler's_method" class="mw-redirect" title="Euler's method">Euler's method</a> or the <a href="Runge%E2%80%93Kutta_methods" title="Runge–Kutta methods">Runge</a><a href="Euler%E2%80%93Bernoulli_beam_theory" title="Euler–Bernoulli beam theory">–</a>Kutta method.
</p><p>In the second step above, a global system of equations is generated from the element equations by transforming coordinates from the subdomains' local nodes to the domain's global nodes. This spatial transformation includes appropriate <a href="Transformation_matrix" title="Transformation matrix">orientation adjustments</a> as applied in relation to the reference <a href="Coordinate_system" title="Coordinate system">coordinate system</a>. The process is often carried out using FEM software with <a href="Coordinates" class="mw-redirect" title="Coordinates">coordinate</a> data generated from the subdomains.
</p><p>The practical application of FEM is known as finite element analysis (FEA). FEA, as applied in <a href="Engineering" title="Engineering">engineering</a>, is a computational tool for performing <a href="Engineering_analysis" title="Engineering analysis">engineering analysis</a>. It includes the use of <a href="Mesh_generation" title="Mesh generation">mesh generation</a> techniques for dividing a <a href="Complex_system" title="Complex system">complex problem</a> into smaller elements, as well as the use of software coded with a FEM algorithm. When applying FEA, the complex problem is usually a physical system with the underlying <a href="Physics" title="Physics">physics</a>, such as the <a href="Euler%E2%80%93Bernoulli_beam_theory" title="Euler–Bernoulli beam theory">Euler–Bernoulli beam equation</a>, the <a href="Heat_equation" title="Heat equation">heat equation</a>, or the <a href="Navier-Stokes_equations" class="mw-redirect" title="Navier-Stokes equations">Navier</a><a href="Euler%E2%80%93Bernoulli_beam_theory" title="Euler–Bernoulli beam theory">–</a>Stokes equations, expressed in either PDEs or <a href="Integral_equation" title="Integral equation">integral equations</a>, while the divided, smaller elements of the complex problem represent different areas in the physical system.
</p><p>FEA may be used for analyzing problems over complicated domains (e.g., cars and oil pipelines) when the domain changes (e.g., during a solid-state reaction with a moving boundary), when the desired precision varies over the entire domain, or when the solution lacks smoothness. FEA simulations provide a valuable resource, as they remove multiple instances of creating and testing complex prototypes for various high-fidelity situations.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> For example, in a frontal crash simulation, it is possible to increase prediction accuracy in important areas, like the front of the car, and reduce it in the rear of the car, thus reducing the cost of the simulation. Another example would be in <a href="Numerical_weather_prediction" title="Numerical weather prediction">numerical weather prediction</a>, where it is more important to have accurate predictions over developing highly nonlinear phenomena, such as <a href="Tropical_cyclone" title="Tropical cyclone">tropical cyclones</a> in the atmosphere or <a href="Eddy_(fluid_dynamics)" title="Eddy (fluid dynamics)">eddies</a> in the ocean, rather than relatively calm areas.
</p><p>A clear, detailed, and practical presentation of this approach can be found in the textbook <i>The Finite Element Method for Engineers</i>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>
<p>While it is difficult to quote the date of the invention of FEM, the method originated from the need to solve complex <a href="Elasticity_(physics)" title="Elasticity (physics)">elasticity</a> and <a href="Structural_analysis" title="Structural analysis">structural analysis</a> problems in <a href="Civil_engineering" title="Civil engineering">civil</a> and <a href="Aeronautical_engineering" class="mw-redirect" title="Aeronautical engineering">aeronautical engineering</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Its development can be traced back to work by <a href="Alexander_Hrennikoff" title="Alexander Hrennikoff">Alexander Hrennikoff</a><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> and <a href="Richard_Courant" title="Richard Courant">Richard Courant</a><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> in the early 1940s. Another pioneer was <a href="Ioannis_Argyris" class="mw-redirect" title="Ioannis Argyris">Ioannis Argyris</a>. In the USSR, the introduction of the practical application of FEM is usually connected with Leonard Oganesyan.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> It was also independently rediscovered in China by <a href="Feng_Kang" title="Feng Kang">Feng Kang</a> in the late 1950s and early 1960s, based on the computations of dam constructions, where it was called the "<a href="Finite_difference_method" title="Finite difference method">finite difference method</a>" based on variation principles. Although the approaches used by these pioneers are different, they share one essential characteristic: the <a href="Polygon_mesh" title="Polygon mesh">mesh</a> <a href="Discretization" title="Discretization">discretization</a> of a continuous domain into a set of discrete sub-domains, usually called elements.
</p><p>Hrennikoff's work discretizes the domain by using a <a href="Lattice_(group)" title="Lattice (group)">lattice</a> analogy, while Courant's approach divides the domain into finite triangular sub-regions to solve <a href="Partial_differential_equation#Linear_equations_of_second_order" title="Partial differential equation">second-order</a> <a href="Elliptic_partial_differential_equation" title="Elliptic partial differential equation">elliptic partial differential equations</a> that arise from the problem of the <a href="Torsion_(mechanics)" title="Torsion (mechanics)">torsion</a> of a <a href="Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">cylinder</a>. Courant's contribution was evolutionary, drawing on a large body of earlier results for PDEs developed by <a href="John_William_Strutt%2C_3rd_Baron_Rayleigh" class="mw-redirect" title="John William Strutt, 3rd Baron Rayleigh">Lord Rayleigh</a>, <a href="Walther_Ritz" title="Walther Ritz">Walther Ritz</a>, and <a href="Boris_Galerkin" title="Boris Galerkin">Boris Galerkin</a>.
</p><p>The application of FEM gained momentum in the 1960s and 1970s due to the developments of <a href="John_Argyris" title="John Argyris">J. H. Argyris</a> and his co-workers at the <a href="University_of_Stuttgart" title="University of Stuttgart">University of Stuttgart</a>; <a href="Ray_W._Clough" class="mw-redirect" title="Ray W. Clough">R. W. Clough</a> and his co-workers at <a href="University_of_California%2C_Berkeley" title="University of California, Berkeley">University of California Berkeley</a>; <a href="Olgierd_Zienkiewicz" title="Olgierd Zienkiewicz">O. C. Zienkiewicz</a> and his co-workers <a href="Ernest_Hinton" title="Ernest Hinton">Ernest Hinton</a>, <a href="Bruce_Irons_(engineer)" title="Bruce Irons (engineer)">Bruce Irons</a>,<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> and others at <a href="Swansea_University" title="Swansea University">Swansea University</a>; <a href="Philippe_G._Ciarlet" title="Philippe G. Ciarlet">Philippe G. Ciarlet</a> at the University of <a href="Pierre-and-Marie-Curie_University" class="mw-redirect" title="Pierre-and-Marie-Curie University">Paris 6</a>; and <a href="Richard_H._Gallagher" title="Richard H. Gallagher">Richard Gallagher</a> and his co-workers at <a href="Cornell_University" title="Cornell University">Cornell University</a>. During this period, additional impetus was provided by the available open-source FEM programs. NASA sponsored the original version of <a href="NASTRAN" class="mw-redirect" title="NASTRAN">NASTRAN</a>. University of California Berkeley made the finite element programs SAP IV<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> and, later, <a href="OpenSees" title="OpenSees">OpenSees</a> widely available. In Norway, the ship classification society Det Norske Veritas (now <a href="DNV_GL" class="mw-redirect" title="DNV GL">DNV GL</a>) developed <a href="SESAM_(FEM)" class="mw-redirect" title="SESAM (FEM)">Sesam</a> in 1969 for use in the analysis of ships.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> A rigorous mathematical basis for FEM was provided in 1973 with a publication by <a href="Gilbert_Strang" title="Gilbert Strang">Gilbert Strang</a> and <a href="George_Fix" title="George Fix">George Fix</a>.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> The method has since been generalized for the <a href="Numerical_analysis" title="Numerical analysis">numerical modeling</a> of physical systems in a wide variety of <a href="Engineering" title="Engineering">engineering</a> disciplines, such as <a href="Electromagnetism" title="Electromagnetism">electromagnetism</a>, <a href="Heat_transfer" title="Heat transfer">heat transfer</a>, and <a href="Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a>.<sup id="cite_ref-ZienkiewiczTaylor2013_14-0" class="reference"><a href="#cite_note-ZienkiewiczTaylor2013-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Technical_discussion">Technical discussion</h2></div>
<div class="mw-heading mw-heading3"><h3 id="The_structure_of_finite_element_methods">The structure of finite element methods</h3></div>
<p>A finite element method is characterized by a <a href="Calculus_of_variations" title="Calculus of variations">variational formulation</a>, a discretization strategy, one or more solution algorithms, and post-processing procedures.
</p><p>Examples of the variational formulation are the <a href="Galerkin_method" title="Galerkin method">Galerkin method</a>, the discontinuous Galerkin method, mixed methods, etc.
</p><p>A discretization strategy is understood to mean a clearly defined set of procedures that cover (a) the creation of finite element meshes, (b) the definition of basis function on reference elements (also called shape functions), and (c) the mapping of reference elements onto the elements of the mesh. Examples of discretization strategies are the h-version, <a href="P-FEM" title="P-FEM">p-version</a>, <a href="Hp-FEM" title="Hp-FEM">hp-version</a>, <a href="Extended_finite_element_method" title="Extended finite element method">x-FEM</a>, <a href="Isogeometric_analysis" title="Isogeometric analysis">isogeometric analysis</a>, etc. Each discretization strategy has certain advantages and disadvantages. A reasonable criterion in selecting a discretization strategy is to realize nearly optimal performance for the broadest set of mathematical models in a particular model class.
</p><p>Various numerical solution algorithms can be classified into two broad categories; direct and iterative solvers. These algorithms are designed to exploit the sparsity of matrices that depend on the variational formulation and discretization strategy choices.
</p><p>Post-processing procedures are designed to extract the data of interest from a finite element solution. To meet the requirements of solution verification, postprocessors need to provide for <i>a posteriori</i> error estimation in terms of the quantities of interest. When the errors of approximation are larger than what is considered acceptable, then the discretization has to be changed either by an automated adaptive process or by the action of the analyst. Some very efficient postprocessors provide for the realization of <a href="Superconvergence" title="Superconvergence">superconvergence</a>.
</p>
<div style="clear:both;" class=""></div>
<div class="mw-heading mw-heading3"><h3 id="Illustrative_problems_P1_and_P2">Illustrative problems P1 and P2</h3></div>
<p>The following two problems demonstrate the finite element method.
</p><p>P1 is a one-dimensional problem
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<annotation encoding="application/x-tex">{\displaystyle u''}</annotation>
</semantics>
</math></span><img src="./75057c7f0c536c8aa2f1df42cfddcfd96985c4f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.467ex; height:2.509ex;" alt="{\displaystyle u''}" loading="lazy"></span> is the second derivative of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>u</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle u}</annotation>
</semantics>
</math></span><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"></span> with respect to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
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</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>.
</p><p>P2 is a two-dimensional problem (<a href="Dirichlet_problem" title="Dirichlet problem">Dirichlet problem</a>)
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{P2 }}:{\begin{cases}u_{xx}(x,y)+u_{yy}(x,y)=f(x,y)&amp;{\text{ in }}\Omega ,\\u=0&amp;{\text{ on }}\partial \Omega ,\end{cases}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtext>P2&nbsp;</mtext>
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<mo>:</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>{</mo>
<mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false">
<mtr>
<mtd>
<msub>
<mi>u</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
<mi>x</mi>
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<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<msub>
<mi>u</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
<mi>y</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>&nbsp;in&nbsp;</mtext>
</mrow>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo>,</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>u</mi>
<mo>=</mo>
<mn>0</mn>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>&nbsp;on&nbsp;</mtext>
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<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo>,</mo>
</mtd>
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<mo fence="true" stretchy="true" symmetric="true"></mo>
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</mrow>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\text{P2 }}:{\begin{cases}u_{xx}(x,y)+u_{yy}(x,y)=f(x,y)&amp;{\text{ in }}\Omega ,\\u=0&amp;{\text{ on }}\partial \Omega ,\end{cases}}}</annotation>
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</math></span></span>
</p><p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>
</semantics>
</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> is a connected open region in the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation>
</semantics>
</math></span><img src="./41cf50e4a314ca8e2c30964baa8d26e5be7a9386.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}" loading="lazy"></span> plane whose boundary <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial \Omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle \partial \Omega }</annotation>
</semantics>
</math></span><img src="./16feddaad462c2a1c9efdaeee062a0484a023fde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.996ex; height:2.176ex;" alt="{\displaystyle \partial \Omega }" loading="lazy"></span> is nice (e.g., a <a href="Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifold</a> or a <a href="Polygon" title="Polygon">polygon</a>), and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{xx}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>u</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
<mi>x</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u_{xx}}</annotation>
</semantics>
</math></span><img src="./c972226aadbf63510b7610318e457526e685b681.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.442ex; height:2.009ex;" alt="{\displaystyle u_{xx}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{yy}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>u</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
<mi>y</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u_{yy}}</annotation>
</semantics>
</math></span><img src="./6a5547e142f37adaf3dc62a0a9ac8e312bbf18e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.196ex; height:2.343ex;" alt="{\displaystyle u_{yy}}" loading="lazy"></span> denote the second derivatives with respect to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y}</annotation>
</semantics>
</math></span><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span>, respectively.
</p><p>The problem P1 can be solved directly by computing <a href="Antiderivative" title="Antiderivative">antiderivatives</a>. However, this method of solving the <a href="Boundary_value_problem" title="Boundary value problem">boundary value problem</a> (BVP) works only when there is one spatial dimension. It does not generalize to higher-dimensional problems or problems like <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u+V''=f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>u</mi>
<mo>+</mo>
<msup>
<mi>V</mi>
<mo>″</mo>
</msup>
<mo>=</mo>
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u+V''=f}</annotation>
</semantics>
</math></span><img src="./9b54f505b218da7280f8d50896868885e7421f10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.601ex; height:2.843ex;" alt="{\displaystyle u+V''=f}" loading="lazy"></span>. For this reason, we will develop the finite element method for P1 and outline its generalization to P2.
</p><p>Our explanation will proceed in two steps, which mirror two essential steps one must take to solve a boundary value problem (BVP) using the FEM.
</p>
<ul><li>In the first step, one rephrases the original BVP in its weak form. Little to no computation is usually required for this step. The transformation is done by hand on paper.</li>
<li>The second step is discretization, where the weak form is discretized in a finite-dimensional space.</li></ul>
<p>After this second step, we have concrete formulae for a large but finite-dimensional linear problem whose solution will approximately solve the original BVP. This finite-dimensional problem is then implemented on a <a href="Computer" title="Computer">computer</a>.
</p>
<div class="mw-heading mw-heading3"><h3 id="Weak_formulation">Weak formulation</h3></div>
<p>The first step is to convert P1 and P2 into their equivalent <a href="Weak_formulation" title="Weak formulation">weak formulations</a>.
</p>
<div class="mw-heading mw-heading4"><h4 id="The_weak_form_of_P1">The weak form of P1</h4></div>
<p>If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>u</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle u}</annotation>
</semantics>
</math></span><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"></span> solves P1, then for any smooth function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>v</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v}</annotation>
</semantics>
</math></span><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"></span> that satisfies the displacement boundary conditions, i.e. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>v</mi>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v=0}</annotation>
</semantics>
</math></span><img src="./ba3d414a23bf4ecfa36cdd039241efc60a5bd9e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.389ex; height:2.176ex;" alt="{\displaystyle v=0}" loading="lazy"></span> at <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x=0}</annotation>
</semantics>
</math></span><img src="./953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x=1}</annotation>
</semantics>
</math></span><img src="./ee42176e76ae6b56d68c42ced807e08b962a2b54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=1}" loading="lazy"></span>, we have
</p>
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</style><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{1}f(x)v(x)\,dx=\int _{0}^{1}u''(x)v(x)\,dx.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<mi>v</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mo>=</mo>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<mrow class="MJX-TeXAtom-ORD">
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<mi>x</mi>
<mo stretchy="false">)</mo>
<mi>v</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{1}f(x)v(x)\,dx=\int _{0}^{1}u''(x)v(x)\,dx.}</annotation>
</semantics>
</math></span></span></td> <td></td> <td class="nowrap"><span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span></td></tr></tbody></table>
<p>Conversely, if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>u</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u}</annotation>
</semantics>
</math></span><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"></span> with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(0)=u(1)=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>u</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>u</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u(0)=u(1)=0}</annotation>
</semantics>
</math></span><img src="./e1a34dfe771c1ceeeddd0fd34558f2e0777de5d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.962ex; height:2.843ex;" alt="{\displaystyle u(0)=u(1)=0}" loading="lazy"></span> satisfies (1) for every smooth function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>v</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v(x)}</annotation>
</semantics>
</math></span><img src="./b371a381e15c71d8fc4ec43cf14b156f02a0d35a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.267ex; height:2.843ex;" alt="{\displaystyle v(x)}" loading="lazy"></span> then one may show that this <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>u</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u}</annotation>
</semantics>
</math></span><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"></span> will solve P1. The proof is easier for twice continuously differentiable <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>u</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u}</annotation>
</semantics>
</math></span><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"></span> (<a href="Mean_value_theorem" title="Mean value theorem">mean value theorem</a>) but may be proved in a <a href="Distribution_(mathematics)" title="Distribution (mathematics)">distributional</a> sense as well.
</p><p>We define a new operator or map <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (u,v)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<mi>u</mi>
<mo>,</mo>
<mi>v</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi (u,v)}</annotation>
</semantics>
</math></span><img src="./857f5ddba44084b873ec4c89532a69db8350e0ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.686ex; height:2.843ex;" alt="{\displaystyle \phi (u,v)}" loading="lazy"></span> by using <a href="Integration_by_parts" title="Integration by parts">integration by parts</a> on the right-hand-side of (1):
</p>
<table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\int _{0}^{1}f(x)v(x)\,dx&amp;=\int _{0}^{1}u''(x)v(x)\,dx\\&amp;=u'(x)v(x)|_{0}^{1}-\int _{0}^{1}u'(x)v'(x)\,dx\\&amp;=-\int _{0}^{1}u'(x)v'(x)\,dx\equiv -\phi (u,v),\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msubsup>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mi>v</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msubsup>
<msup>
<mi>u</mi>
<mo>″</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mi>v</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<msup>
<mi>u</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mi>v</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<msubsup>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msubsup>
<mo>−<!-- − --></mo>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msubsup>
<msup>
<mi>u</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<msup>
<mi>v</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msubsup>
<msup>
<mi>u</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<msup>
<mi>v</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mo>≡<!-- ≡ --></mo>
<mo>−<!-- − --></mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<mi>u</mi>
<mo>,</mo>
<mi>v</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int _{0}^{1}f(x)v(x)\,dx&amp;=\int _{0}^{1}u''(x)v(x)\,dx\\&amp;=u'(x)v(x)|_{0}^{1}-\int _{0}^{1}u'(x)v'(x)\,dx\\&amp;=-\int _{0}^{1}u'(x)v'(x)\,dx\equiv -\phi (u,v),\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./aca8e65b05e6fd22bf65e79f8abb32de0537a59e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.74ex; margin-bottom: -0.265ex; width:50.75ex; height:19.176ex;" alt="{\displaystyle {\begin{aligned}\int _{0}^{1}f(x)v(x)\,dx&amp;=\int _{0}^{1}u''(x)v(x)\,dx\\&amp;=u'(x)v(x)|_{0}^{1}-\int _{0}^{1}u'(x)v'(x)\,dx\\&amp;=-\int _{0}^{1}u'(x)v'(x)\,dx\equiv -\phi (u,v),\end{aligned}}}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span></td></tr></tbody></table>
<p>where we have used the assumption that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(0)=v(1)=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>v</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>v</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v(0)=v(1)=0}</annotation>
</semantics>
</math></span><img src="./918328090764fcd5813a5c42ee1fd5c06565e4cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.558ex; height:2.843ex;" alt="{\displaystyle v(0)=v(1)=0}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading4"><h4 id="The_weak_form_of_P2">The weak form of P2</h4></div>
<p>If we integrate by parts using a form of <a href="Green's_identities" title="Green's identities">Green's identities</a>, we see that if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>u</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u}</annotation>
</semantics>
</math></span><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"></span> solves P2, then we may define <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (u,v)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<mi>u</mi>
<mo>,</mo>
<mi>v</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi (u,v)}</annotation>
</semantics>
</math></span><img src="./857f5ddba44084b873ec4c89532a69db8350e0ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.686ex; height:2.843ex;" alt="{\displaystyle \phi (u,v)}" loading="lazy"></span> for any <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>v</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v}</annotation>
</semantics>
</math></span><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"></span> by
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\Omega }fv\,ds=-\int _{\Omega }\nabla u\cdot \nabla v\,ds\equiv -\phi (u,v),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
</mrow>
</msub>
<mi>f</mi>
<mi>v</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>s</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
</mrow>
</msub>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mi>u</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mi>v</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>s</mi>
<mo>≡<!-- ≡ --></mo>
<mo>−<!-- − --></mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<mi>u</mi>
<mo>,</mo>
<mi>v</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \int _{\Omega }fv\,ds=-\int _{\Omega }\nabla u\cdot \nabla v\,ds\equiv -\phi (u,v),}</annotation>
</semantics>
</math></span></span>
</p><p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∇<!-- ∇ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla }</annotation>
</semantics>
</math></span><img src="./a3d0e93b78c50237f9ea83d027e4ebbdaef354b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \nabla }" loading="lazy"></span> denotes the <a href="Gradient" title="Gradient">gradient</a> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>⋅<!-- ⋅ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation>
</semantics>
</math></span><img src="./ba2c023bad1bd39ed49080f729cbf26bc448c9ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }" loading="lazy"></span> denotes the <a href="Dot_product" title="Dot product">dot product</a> in the two-dimensional plane. Once more <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\!\phi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mspace width="thinmathspace"></mspace>
<mspace width="negativethinmathspace"></mspace>
<mi>ϕ<!-- ϕ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \,\!\phi }</annotation>
</semantics>
</math></span><img src="./a53894ae6e1efd0e272291abea516a222ace8a4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \,\!\phi }" loading="lazy"></span> can be turned into an inner product on a suitable space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}^{1}(\Omega )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H_{0}^{1}(\Omega )}</annotation>
</semantics>
</math></span><img src="./830c3c449349317a46b77590a546b7af4a62dc08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.645ex; height:3.176ex;" alt="{\displaystyle H_{0}^{1}(\Omega )}" loading="lazy"></span> of once differentiable functions of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>
</semantics>
</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> that are zero on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial \Omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \partial \Omega }</annotation>
</semantics>
</math></span><img src="./16feddaad462c2a1c9efdaeee062a0484a023fde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.996ex; height:2.176ex;" alt="{\displaystyle \partial \Omega }" loading="lazy"></span>. We have also assumed that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in H_{0}^{1}(\Omega )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>v</mi>
<mo>∈<!-- ∈ --></mo>
<msubsup>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v\in H_{0}^{1}(\Omega )}</annotation>
</semantics>
</math></span><img src="./5b528325e0f9461d9c2710717a5cde8f2c7116f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.613ex; height:3.176ex;" alt="{\displaystyle v\in H_{0}^{1}(\Omega )}" loading="lazy"></span> (see <a href="Sobolev_space" title="Sobolev space">Sobolev spaces</a>). The existence and uniqueness of the solution can also be shown.
</p>
<div class="mw-heading mw-heading4"><h4 id="A_proof_outline_of_the_existence_and_uniqueness_of_the_solution">A proof outline of the existence and uniqueness of the solution</h4></div>
<p>We can loosely think of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}^{1}(0,1)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H_{0}^{1}(0,1)}</annotation>
</semantics>
</math></span><img src="./a7d3d5f81eae348b97a89a1b17d6c84e01544366.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.326ex; height:3.176ex;" alt="{\displaystyle H_{0}^{1}(0,1)}" loading="lazy"></span> to be the <a href="Absolutely_continuous" class="mw-redirect" title="Absolutely continuous">absolutely continuous</a> functions of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation>
</semantics>
</math></span><img src="./c79c6838e423c1ed3c7ea532a56dc9f9dae8290b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,1)}" loading="lazy"></span> that are <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>
</semantics>
</math></span><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"></span> at <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x=0}</annotation>
</semantics>
</math></span><img src="./953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x=1}</annotation>
</semantics>
</math></span><img src="./ee42176e76ae6b56d68c42ced807e08b962a2b54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=1}" loading="lazy"></span> (see <a href="Sobolev_spaces" class="mw-redirect" title="Sobolev spaces">Sobolev spaces</a>). Such functions are (weakly) once differentiable, and it turns out that the symmetric <a href="Bilinear_map" title="Bilinear map">bilinear map</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \!\,\phi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mspace width="negativethinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>ϕ<!-- ϕ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \!\,\phi }</annotation>
</semantics>
</math></span><img src="./1315e3582db5f2a61f20766851bad6f0860e5069.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.387ex; width:1.773ex; height:2.509ex;" alt="{\displaystyle \!\,\phi }" loading="lazy"></span> then defines an <a href="Inner_product" class="mw-redirect" title="Inner product">inner product</a> which turns <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}^{1}(0,1)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H_{0}^{1}(0,1)}</annotation>
</semantics>
</math></span><img src="./a7d3d5f81eae348b97a89a1b17d6c84e01544366.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.326ex; height:3.176ex;" alt="{\displaystyle H_{0}^{1}(0,1)}" loading="lazy"></span> into a <a href="Hilbert_space" title="Hilbert space">Hilbert space</a> (a detailed proof is nontrivial). On the other hand, the left-hand-side <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{1}f(x)v(x)dx}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msubsup>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mi>v</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mi>d</mi>
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{1}f(x)v(x)dx}</annotation>
</semantics>
</math></span><img src="./fca47c7a1b9cf10efac7b342c6dccad88f97a16c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.135ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{1}f(x)v(x)dx}" loading="lazy"></span> is also an inner product, this time on the <a href="Lp_space" title="Lp space">Lp space</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(0,1)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L^{2}(0,1)}</annotation>
</semantics>
</math></span><img src="./3cc63d7b243a51a699bcb6d6cf30b6ca2a4a65a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.805ex; height:3.176ex;" alt="{\displaystyle L^{2}(0,1)}" loading="lazy"></span>. An application of the <a href="Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation theorem</a> for Hilbert spaces shows that there is a unique <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>u</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u}</annotation>
</semantics>
</math></span><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"></span> solving (2) and, therefore, P1. This solution is a-priori only a member of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}^{1}(0,1)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H_{0}^{1}(0,1)}</annotation>
</semantics>
</math></span><img src="./a7d3d5f81eae348b97a89a1b17d6c84e01544366.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.326ex; height:3.176ex;" alt="{\displaystyle H_{0}^{1}(0,1)}" loading="lazy"></span>, but using <a href="Elliptic_operator" title="Elliptic operator">elliptic</a> regularity, will be smooth if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is.
</p>
<div class="mw-heading mw-heading2"><h2 id="Discretization">Discretization</h2></div>

<p>P1 and P2 are ready to be discretized, which leads to a common sub-problem (3). The basic idea is to replace the infinite-dimensional linear problem:
</p>
<dl><dd>Find <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in H_{0}^{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>u</mi>
<mo>∈<!-- ∈ --></mo>
<msubsup>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u\in H_{0}^{1}}</annotation>
</semantics>
</math></span><img src="./13251cb612942c56abf6aa865b546cf5204f0932.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.328ex; height:3.176ex;" alt="{\displaystyle u\in H_{0}^{1}}" loading="lazy"></span> such that</dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall v\in H_{0}^{1},\;-\phi (u,v)=\int fv}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∀<!-- ∀ --></mi>
<mi>v</mi>
<mo>∈<!-- ∈ --></mo>
<msubsup>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msubsup>
<mo>,</mo>
<mspace width="thickmathspace"></mspace>
<mo>−<!-- − --></mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<mi>u</mi>
<mo>,</mo>
<mi>v</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>∫<!-- ∫ --></mo>
<mi>f</mi>
<mi>v</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \forall v\in H_{0}^{1},\;-\phi (u,v)=\int fv}</annotation>
</semantics>
</math></span><img src="./afb0a124f7f851f7ffda822cafc67b8f8f3dccc2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.677ex; height:5.676ex;" alt="{\displaystyle \forall v\in H_{0}^{1},\;-\phi (u,v)=\int fv}" loading="lazy"></span></dd></dl>
<p>with a finite-dimensional version:
</p>
<table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap">Find <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>u</mi>
<mo>∈<!-- ∈ --></mo>
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u\in V}</annotation>
</semantics>
</math></span><img src="./636dd20088dea1139b38b3c04053ccf508bbed8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.958ex; height:2.176ex;" alt="{\displaystyle u\in V}" loading="lazy"></span> such that<br>
<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall v\in V,\;-\phi (u,v)=\int fv}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∀<!-- ∀ --></mi>
<mi>v</mi>
<mo>∈<!-- ∈ --></mo>
<mi>V</mi>
<mo>,</mo>
<mspace width="thickmathspace"></mspace>
<mo>−<!-- − --></mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<mi>u</mi>
<mo>,</mo>
<mi>v</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>∫<!-- ∫ --></mo>
<mi>f</mi>
<mi>v</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \forall v\in V,\;-\phi (u,v)=\int fv}</annotation>
</semantics>
</math></span><img src="./c4c819c7a75457354318c1ce1f3a94b92f42f9b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.307ex; height:5.676ex;" alt="{\displaystyle \forall v\in V,\;-\phi (u,v)=\int fv}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_3" class="reference nourlexpansion" style="font-weight:bold;">3</span></td></tr></tbody></table>
<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
</semantics>
</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> is a finite-dimensional <a href="Linear_subspace" title="Linear subspace">subspace</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}^{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H_{0}^{1}}</annotation>
</semantics>
</math></span><img src="./fd9c9e1684f33b9116efe894347370f5bfc0c8e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.158ex; height:3.176ex;" alt="{\displaystyle H_{0}^{1}}" loading="lazy"></span>. There are many possible choices for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
</semantics>
</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> (one possibility leads to the <a href="Spectral_method" title="Spectral method">spectral method</a>). However, we take <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
</semantics>
</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> as a space of piecewise polynomial functions for the finite element method.
</p>
<div class="mw-heading mw-heading3"><h3 id="For_problem_P1">For problem P1</h3></div>
<p>We take the interval <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation>
</semantics>
</math></span><img src="./c79c6838e423c1ed3c7ea532a56dc9f9dae8290b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,1)}" loading="lazy"></span>, choose <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> values of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=x_{0}<x_{1}<\cdots <x_{n}<x_{n+1}=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
<mo>=</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>&lt;</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>&lt;</mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>&lt;</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>&lt;</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 0=x_{0}&lt;x_{1}&lt;\cdots &lt;x_{n}&lt;x_{n+1}=1}</annotation>
</semantics>
</math></span><img src="./7730812aae7cfa944b21b6e06c10f611c91433d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:35.604ex; height:2.509ex;" alt="{\displaystyle 0=x_{0}<x_{1}<\cdots <x_{n}<x_{n+1}=1}" loading="lazy"></span> and we define <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
</semantics>
</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> by:
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\{v:[0,1]\to \mathbb {R} \;:v{\text{ is continuous, }}v|_{[x_{k},x_{k+1}]}{\text{ is linear for }}k=0,\dots ,n{\text{, and }}v(0)=v(1)=0\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>v</mi>
<mo>:</mo>
<mo stretchy="false">[</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo stretchy="false">]</mo>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mspace width="thickmathspace"></mspace>
<mo>:</mo>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>&nbsp;is continuous,&nbsp;</mtext>
</mrow>
<mi>v</mi>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">[</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">]</mo>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mtext>&nbsp;is linear for&nbsp;</mtext>
</mrow>
<mi>k</mi>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>, and&nbsp;</mtext>
</mrow>
<mi>v</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>v</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V=\{v:[0,1]\to \mathbb {R} \;:v{\text{ is continuous, }}v|_{[x_{k},x_{k+1}]}{\text{ is linear for }}k=0,\dots ,n{\text{, and }}v(0)=v(1)=0\}}</annotation>
</semantics>
</math></span></span>
</p><p>where we define <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{0}=0}</annotation>
</semantics>
</math></span><img src="./9d18a96da37e1748deeb8d4c590dd4ad6629efef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.645ex; height:2.509ex;" alt="{\displaystyle x_{0}=0}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n+1}=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{n+1}=1}</annotation>
</semantics>
</math></span><img src="./094e2edc998bc87ff0e124c456fb9cf757f92141.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.91ex; height:2.509ex;" alt="{\displaystyle x_{n+1}=1}" loading="lazy"></span>. Observe that functions in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
</semantics>
</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> are not differentiable according to the elementary definition of calculus. Indeed, if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>v</mi>
<mo>∈<!-- ∈ --></mo>
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v\in V}</annotation>
</semantics>
</math></span><img src="./99886ebbde63daa0224fb9bf56fa11b3c8a6f4fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.756ex; height:2.176ex;" alt="{\displaystyle v\in V}" loading="lazy"></span> then the derivative is typically not defined at any <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x_{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>=</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x=x_{k}}</annotation>
</semantics>
</math></span><img src="./2e676498d1003087514f2b5ad431f1e73945c565.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.847ex; height:2.009ex;" alt="{\displaystyle x=x_{k}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1,\ldots ,n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k=1,\ldots ,n}</annotation>
</semantics>
</math></span><img src="./02703686f808b37fedb436806fa72ca3522e22de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.045ex; height:2.509ex;" alt="{\displaystyle k=1,\ldots ,n}" loading="lazy"></span>. However, the derivative exists at every other value of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>, and one can use this derivative for <a href="Integration_by_parts" title="Integration by parts">integration by parts</a>.
</p>

<div class="mw-heading mw-heading3"><h3 id="For_problem_P2">For problem P2</h3></div>
<p>We need <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
</semantics>
</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> to be a set of functions of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>
</semantics>
</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span>. In the figure on the right, we have illustrated a <a href="Polygon_triangulation" title="Polygon triangulation">triangulation</a> of a 15-sided <a href="Polygon" title="Polygon">polygonal</a> region <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>
</semantics>
</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> in the plane (below), and a <a href="Piecewise_linear_function" title="Piecewise linear function">piecewise linear function</a> (above, in color) of this polygon which is linear on each triangle of the triangulation; the space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
</semantics>
</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> would consist of functions that are linear on each triangle of the chosen triangulation.
</p><p>One hopes that as the underlying triangular mesh becomes finer and finer, the solution of the discrete problem (3) will, in some sense, converge to the solution of the original boundary value problem P2. To measure this mesh fineness, the triangulation is indexed by a real-valued parameter <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h>0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
<mo>&gt;</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h&gt;0}</annotation>
</semantics>
</math></span><img src="./cbddb7a5cca6170575e4e73e769fbb434c2a3d71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.6ex; height:2.176ex;" alt="{\displaystyle h>0}" loading="lazy"></span> which one takes to be very small. This parameter will be related to the largest or average triangle size in the triangulation. As we refine the triangulation, the space of piecewise linear functions <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
</semantics>
</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> must also change with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h}</annotation>
</semantics>
</math></span><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"></span>. For this reason, one often reads <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{h}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>V</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>h</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V_{h}}</annotation>
</semantics>
</math></span><img src="./652f5cdfa49da86f90fa98f1ab5c47a3384f1464.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.534ex; height:2.509ex;" alt="{\displaystyle V_{h}}" loading="lazy"></span> instead of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
</semantics>
</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> in the literature. Since we do not perform such an analysis, we will not use this notation.
</p>
<div class="mw-heading mw-heading3"><h3 id="Choosing_a_basis">Choosing a basis</h3></div>
<div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:204px;max-width:204px"><div class="trow"><div class="theader" style="text-align:center">Interpolation of a <a href="Bessel_function" title="Bessel function">Bessel function</a></div></div><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"></span></div><div class="thumbcaption text-align-left">16 scaled and shifted triangular basis functions (colors) used to reconstruct a zeroeth order Bessel function <i>J</i><sub><i>0</i></sub> (black)</div></div></div><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"></span></div><div class="thumbcaption text-align-left">The linear combination of basis functions (yellow) reproduces <i>J</i><sub><i>0</i></sub> (black) to any desired accuracy.</div></div></div></div></div>
<p>To complete the discretization, we must select a <a href="Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
</semantics>
</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span>. In the one-dimensional case, for each control point <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{k}}</annotation>
</semantics>
</math></span><img src="./6d2b88c64c76a03611549fb9b4cf4ed060b56002.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.009ex;" alt="{\displaystyle x_{k}}" loading="lazy"></span> we will choose the piecewise linear function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v_{k}}</annotation>
</semantics>
</math></span><img src="./d142b4083872eb72f81c1e20fd2c91d02b4a9838.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.009ex;" alt="{\displaystyle v_{k}}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
</semantics>
</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> whose value is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>
</semantics>
</math></span><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"></span> at <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{k}}</annotation>
</semantics>
</math></span><img src="./6d2b88c64c76a03611549fb9b4cf4ed060b56002.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.009ex;" alt="{\displaystyle x_{k}}" loading="lazy"></span> and zero at every <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j},\;j\neq k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>,</mo>
<mspace width="thickmathspace"></mspace>
<mi>j</mi>
<mo>≠<!-- ≠ --></mo>
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{j},\;j\neq k}</annotation>
</semantics>
</math></span><img src="./7f729084d28d94a3aa9f08ed2e4cdcf7e4f84b9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.186ex; height:2.843ex;" alt="{\displaystyle x_{j},\;j\neq k}" loading="lazy"></span>, i.e.,
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{k}(x)={\begin{cases}{x-x_{k-1} \over x_{k}\,-x_{k-1}}&amp;{\text{ if }}x\in [x_{k-1},x_{k}],\\{x_{k+1}\,-x \over x_{k+1}\,-x_{k}}&amp;{\text{ if }}x\in [x_{k},x_{k+1}],\\0&amp;{\text{ otherwise}},\end{cases}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>{</mo>
<mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false">
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>x</mi>
<mo>−<!-- − --></mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msub>
</mrow>
<mrow>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mspace width="thinmathspace"></mspace>
<mo>−<!-- − --></mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>&nbsp;if&nbsp;</mtext>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mo stretchy="false">[</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo stretchy="false">]</mo>
<mo>,</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mspace width="thinmathspace"></mspace>
<mo>−<!-- − --></mo>
<mi>x</mi>
</mrow>
<mrow>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mspace width="thinmathspace"></mspace>
<mo>−<!-- − --></mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>&nbsp;if&nbsp;</mtext>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mo stretchy="false">[</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">]</mo>
<mo>,</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>&nbsp;otherwise</mtext>
</mrow>
<mo>,</mo>
</mtd>
</mtr>
</mtable>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v_{k}(x)={\begin{cases}{x-x_{k-1} \over x_{k}\,-x_{k-1}}&amp;{\text{ if }}x\in [x_{k-1},x_{k}],\\{x_{k+1}\,-x \over x_{k+1}\,-x_{k}}&amp;{\text{ if }}x\in [x_{k},x_{k+1}],\\0&amp;{\text{ otherwise}},\end{cases}}}</annotation>
</semantics>
</math></span></span>
</p><p>for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1,\dots ,n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k=1,\dots ,n}</annotation>
</semantics>
</math></span><img src="./4cea3d30a653b96d33958125a11c37add9d66a92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.045ex; height:2.509ex;" alt="{\displaystyle k=1,\dots ,n}" loading="lazy"></span>; this basis is a shifted and scaled <a href="Tent_function" class="mw-redirect" title="Tent function">tent function</a>. For the two-dimensional case, we choose again one basis function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v_{k}}</annotation>
</semantics>
</math></span><img src="./d142b4083872eb72f81c1e20fd2c91d02b4a9838.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.009ex;" alt="{\displaystyle v_{k}}" loading="lazy"></span> per vertex <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{k}}</annotation>
</semantics>
</math></span><img src="./6d2b88c64c76a03611549fb9b4cf4ed060b56002.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.009ex;" alt="{\displaystyle x_{k}}" loading="lazy"></span> of the triangulation of the planar region <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>
</semantics>
</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span>. The function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v_{k}}</annotation>
</semantics>
</math></span><img src="./d142b4083872eb72f81c1e20fd2c91d02b4a9838.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.009ex;" alt="{\displaystyle v_{k}}" loading="lazy"></span> is the unique function of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
</semantics>
</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> whose value is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>
</semantics>
</math></span><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"></span> at <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{k}}</annotation>
</semantics>
</math></span><img src="./6d2b88c64c76a03611549fb9b4cf4ed060b56002.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.009ex;" alt="{\displaystyle x_{k}}" loading="lazy"></span> and zero at every <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j},\;j\neq k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>,</mo>
<mspace width="thickmathspace"></mspace>
<mi>j</mi>
<mo>≠<!-- ≠ --></mo>
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{j},\;j\neq k}</annotation>
</semantics>
</math></span><img src="./7f729084d28d94a3aa9f08ed2e4cdcf7e4f84b9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.186ex; height:2.843ex;" alt="{\displaystyle x_{j},\;j\neq k}" loading="lazy"></span>.
</p><p>Depending on the author, the word "element" in the "finite element method" refers to the domain's triangles, the piecewise linear basis function, or both. So, for instance, an author interested in curved domains might replace the triangles with curved primitives and so might describe the elements as being curvilinear. On the other hand, some authors replace "piecewise linear" with "piecewise quadratic" or even "piecewise polynomial". The author might then say "higher order element" instead of "higher degree polynomial". The finite element method is not restricted to triangles (tetrahedra in 3-d or higher-order simplexes in multidimensional spaces). Still, it can be defined on quadrilateral subdomains (hexahedra, prisms, or pyramids in 3-d, and so on). Higher-order shapes (curvilinear elements) can be defined with polynomial and even non-polynomial shapes (e.g., ellipse or circle).
</p><p>Examples of methods that use higher degree piecewise polynomial basis functions are the <a href="Hp-FEM" title="Hp-FEM">hp-FEM</a> and <a href="Spectral_element_method" title="Spectral element method">spectral FEM</a>.
</p><p>More advanced implementations (adaptive finite element methods) utilize a method to assess the quality of the results (based on error estimation theory) and modify the mesh during the solution aiming to achieve an approximate solution within some bounds from the exact solution of the continuum problem. Mesh adaptivity may utilize various techniques; the most popular are:
</p>
<ul><li>moving nodes (r-adaptivity)</li>
<li>refining (and unrefined) elements (h-adaptivity)</li>
<li>changing order of base functions (p-adaptivity)</li>
<li>combinations of the above (<a href="Hp-FEM" title="Hp-FEM">hp-adaptivity</a>).</li></ul>
<div class="mw-heading mw-heading3"><h3 id="Small_support_of_the_basis">Small support of the basis</h3></div>



<p>The primary advantage of this choice of basis is that the inner products
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle v_{j},v_{k}\rangle =\int _{0}^{1}v_{j}v_{k}\,dx}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
<mo>=</mo>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msubsup>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \langle v_{j},v_{k}\rangle =\int _{0}^{1}v_{j}v_{k}\,dx}</annotation>
</semantics>
</math></span></span>
and
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (v_{j},v_{k})=\int _{0}^{1}v_{j}'v_{k}'\,dx}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msubsup>
<msubsup>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
<mo>′</mo>
</msubsup>
<msubsup>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
<mo>′</mo>
</msubsup>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi (v_{j},v_{k})=\int _{0}^{1}v_{j}'v_{k}'\,dx}</annotation>
</semantics>
</math></span></span>
will be zero for almost all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j,k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle j,k}</annotation>
</semantics>
</math></span><img src="./d23e18a251a10a993e66d41e8dbcaf858ba4fa5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:3.23ex; height:2.509ex;" alt="{\displaystyle j,k}" loading="lazy"></span>.
(The matrix containing <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle v_{j},v_{k}\rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \langle v_{j},v_{k}\rangle }</annotation>
</semantics>
</math></span><img src="./d2a4c45ed6c6dc82ecae85127175860afb2f67bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.097ex; height:3.009ex;" alt="{\displaystyle \langle v_{j},v_{k}\rangle }" loading="lazy"></span> in the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (j,k)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (j,k)}</annotation>
</semantics>
</math></span><img src="./6d001cfb64ce123bb1a5ddf9047cd949d7753145.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.013ex; height:2.843ex;" alt="{\displaystyle (j,k)}" loading="lazy"></span> location is known as the <a href="Gramian_matrix" class="mw-redirect" title="Gramian matrix">Gramian matrix</a>.)
In the one dimensional case, the <a href="Support_(mathematics)" title="Support (mathematics)">support</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v_{k}}</annotation>
</semantics>
</math></span><img src="./d142b4083872eb72f81c1e20fd2c91d02b4a9838.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.009ex;" alt="{\displaystyle v_{k}}" loading="lazy"></span> is the interval <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x_{k-1},x_{k+1}]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [x_{k-1},x_{k+1}]}</annotation>
</semantics>
</math></span><img src="./e9ab905e7068db95aeddfd9f0fa55af0e0363843.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.365ex; height:2.843ex;" alt="{\displaystyle [x_{k-1},x_{k+1}]}" loading="lazy"></span>. Hence, the integrands of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle v_{j},v_{k}\rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \langle v_{j},v_{k}\rangle }</annotation>
</semantics>
</math></span><img src="./d2a4c45ed6c6dc82ecae85127175860afb2f67bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.097ex; height:3.009ex;" alt="{\displaystyle \langle v_{j},v_{k}\rangle }" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (v_{j},v_{k})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi (v_{j},v_{k})}</annotation>
</semantics>
</math></span><img src="./43a01aec768783146ac1c1b74d1aa362ca0065af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.482ex; height:3.009ex;" alt="{\displaystyle \phi (v_{j},v_{k})}" loading="lazy"></span> are identically zero whenever <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |j-k|>1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>j</mi>
<mo>−<!-- − --></mo>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>&gt;</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |j-k|&gt;1}</annotation>
</semantics>
</math></span><img src="./b1fc83292a5fed7bcc2e8f170da5ec530ef32aa1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.564ex; height:2.843ex;" alt="{\displaystyle |j-k|>1}" loading="lazy"></span>.
</p><p>Similarly, in the planar case, if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{j}}</annotation>
</semantics>
</math></span><img src="./5db47cb3d2f9496205a17a6856c91c1d3d363ccd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.239ex; height:2.343ex;" alt="{\displaystyle x_{j}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{k}}</annotation>
</semantics>
</math></span><img src="./6d2b88c64c76a03611549fb9b4cf4ed060b56002.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.009ex;" alt="{\displaystyle x_{k}}" loading="lazy"></span> do not share an edge of the triangulation, then the integrals
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\Omega }v_{j}v_{k}\,ds}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
</mrow>
</msub>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>s</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \int _{\Omega }v_{j}v_{k}\,ds}</annotation>
</semantics>
</math></span></span>
and
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\Omega }\nabla v_{j}\cdot \nabla v_{k}\,ds}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
</mrow>
</msub>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>s</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \int _{\Omega }\nabla v_{j}\cdot \nabla v_{k}\,ds}</annotation>
</semantics>
</math></span></span>
are both zero.
</p>
<div class="mw-heading mw-heading3"><h3 id="Matrix_form_of_the_problem">Matrix form of the problem</h3></div>
<p>If we write <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(x)=\sum _{k=1}^{n}u_{k}v_{k}(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>u</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<msub>
<mi>u</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u(x)=\sum _{k=1}^{n}u_{k}v_{k}(x)}</annotation>
</semantics>
</math></span><img src="./d048e6042ecb9920d5d78693f9cd0e162cc4567e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.083ex; height:6.843ex;" alt="{\displaystyle u(x)=\sum _{k=1}^{n}u_{k}v_{k}(x)}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\sum _{k=1}^{n}f_{k}v_{k}(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x)=\sum _{k=1}^{n}f_{k}v_{k}(x)}</annotation>
</semantics>
</math></span><img src="./8755493ff0ad90921dcd691ea6bbf55810f96aa8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.841ex; height:6.843ex;" alt="{\displaystyle f(x)=\sum _{k=1}^{n}f_{k}v_{k}(x)}" loading="lazy"></span> then problem (3), taking <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(x)=v_{j}(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>v</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v(x)=v_{j}(x)}</annotation>
</semantics>
</math></span><img src="./ffac80797632924fc24db91fa08fe38ed82f3ea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.541ex; height:3.009ex;" alt="{\displaystyle v(x)=v_{j}(x)}" loading="lazy"></span> for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1,\dots ,n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle j=1,\dots ,n}</annotation>
</semantics>
</math></span><img src="./6f9393eaa189f1fb2c747b687b7b8d67640d5f1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:11.819ex; height:2.509ex;" alt="{\displaystyle j=1,\dots ,n}" loading="lazy"></span>, becomes
</p>
<table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\sum _{k=1}^{n}u_{k}\phi (v_{k},v_{j})=\sum _{k=1}^{n}f_{k}\int v_{k}v_{j}dx}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>−<!-- − --></mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<msub>
<mi>u</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>∫<!-- ∫ --></mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mi>d</mi>
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle -\sum _{k=1}^{n}u_{k}\phi (v_{k},v_{j})=\sum _{k=1}^{n}f_{k}\int v_{k}v_{j}dx}</annotation>
</semantics>
</math></span><img src="./52b7db3f81feaa3057a676e7d964092f28dffc7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.674ex; height:6.843ex;" alt="{\displaystyle -\sum _{k=1}^{n}u_{k}\phi (v_{k},v_{j})=\sum _{k=1}^{n}f_{k}\int v_{k}v_{j}dx}" loading="lazy"></span> for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1,\dots ,n.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>n</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle j=1,\dots ,n.}</annotation>
</semantics>
</math></span><img src="./78fbe66f1d761a4c3189e16cfb5459335c8b20c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:12.465ex; height:2.509ex;" alt="{\displaystyle j=1,\dots ,n.}" loading="lazy"></span> </td> <td></td> <td class="nowrap"><span id="math_4" class="reference nourlexpansion" style="font-weight:bold;">4</span></td></tr></tbody></table>
<p>If we denote by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">u</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} }</annotation>
</semantics>
</math></span><img src="./261e20fe101de02a771021d9d4466c0ad3e352d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">f</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} }</annotation>
</semantics>
</math></span><img src="./dc6194e680a4e7c521f2178c50eea302843a852d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.053ex; height:2.176ex;" alt="{\displaystyle \mathbf {f} }" loading="lazy"></span> the column vectors <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u_{1},\dots ,u_{n})^{t}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<msub>
<mi>u</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>u</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (u_{1},\dots ,u_{n})^{t}}</annotation>
</semantics>
</math></span><img src="./7b3a99d74915a3d8c937df712e359fc849c7fb62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.746ex; height:3.009ex;" alt="{\displaystyle (u_{1},\dots ,u_{n})^{t}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f_{1},\dots ,f_{n})^{t}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (f_{1},\dots ,f_{n})^{t}}</annotation>
</semantics>
</math></span><img src="./bbf014eef21cf396342cb9bf18ee343c5ac65552.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.365ex; height:3.009ex;" alt="{\displaystyle (f_{1},\dots ,f_{n})^{t}}" loading="lazy"></span>, and if we let
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=(L_{ij})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L=(L_{ij})}</annotation>
</semantics>
</math></span></span>
and
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=(M_{ij})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M=(M_{ij})}</annotation>
</semantics>
</math></span></span>
be matrices whose entries are
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{ij}=\phi (v_{i},v_{j})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L_{ij}=\phi (v_{i},v_{j})}</annotation>
</semantics>
</math></span></span>
and
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{ij}=\int v_{i}v_{j}dx}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<mo>∫<!-- ∫ --></mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mi>d</mi>
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M_{ij}=\int v_{i}v_{j}dx}</annotation>
</semantics>
</math></span></span>
then we may rephrase (4) as
</p>
<table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -L\mathbf {u} =M\mathbf {f} .}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>−<!-- − --></mo>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">u</mi>
</mrow>
<mo>=</mo>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">f</mi>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle -L\mathbf {u} =M\mathbf {f} .}</annotation>
</semantics>
</math></span></span></td> <td></td> <td class="nowrap"><span id="math_5" class="reference nourlexpansion" style="font-weight:bold;">5</span></td></tr></tbody></table>
<p>It is not necessary to assume <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\sum _{k=1}^{n}f_{k}v_{k}(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x)=\sum _{k=1}^{n}f_{k}v_{k}(x)}</annotation>
</semantics>
</math></span><img src="./8755493ff0ad90921dcd691ea6bbf55810f96aa8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.841ex; height:6.843ex;" alt="{\displaystyle f(x)=\sum _{k=1}^{n}f_{k}v_{k}(x)}" loading="lazy"></span>. For a general function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>
</semantics>
</math></span><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"></span>, problem (3) with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(x)=v_{j}(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>v</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v(x)=v_{j}(x)}</annotation>
</semantics>
</math></span><img src="./ffac80797632924fc24db91fa08fe38ed82f3ea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.541ex; height:3.009ex;" alt="{\displaystyle v(x)=v_{j}(x)}" loading="lazy"></span> for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1,\dots ,n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle j=1,\dots ,n}</annotation>
</semantics>
</math></span><img src="./6f9393eaa189f1fb2c747b687b7b8d67640d5f1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:11.819ex; height:2.509ex;" alt="{\displaystyle j=1,\dots ,n}" loading="lazy"></span> becomes actually simpler, since no matrix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M}</annotation>
</semantics>
</math></span><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> is used,
</p>
<table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -L\mathbf {u} =\mathbf {b} ,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>−<!-- − --></mo>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">u</mi>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">b</mi>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle -L\mathbf {u} =\mathbf {b} ,}</annotation>
</semantics>
</math></span></span></td> <td></td> <td class="nowrap"><span id="math_6" class="reference nourlexpansion" style="font-weight:bold;">6</span></td></tr></tbody></table>
<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} =(b_{1},\dots ,b_{n})^{t}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">b</mi>
</mrow>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {b} =(b_{1},\dots ,b_{n})^{t}}</annotation>
</semantics>
</math></span><img src="./9133ba3635dd43f495d260c45bde8bd61a60090c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.665ex; height:3.009ex;" alt="{\displaystyle \mathbf {b} =(b_{1},\dots ,b_{n})^{t}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{j}=\int fv_{j}dx}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<mo>∫<!-- ∫ --></mo>
<mi>f</mi>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mi>d</mi>
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b_{j}=\int fv_{j}dx}</annotation>
</semantics>
</math></span><img src="./27e3e00504a55ef0b934e99e6f7516cd356a44b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.448ex; height:5.676ex;" alt="{\displaystyle b_{j}=\int fv_{j}dx}" loading="lazy"></span> for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1,\dots ,n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle j=1,\dots ,n}</annotation>
</semantics>
</math></span><img src="./6f9393eaa189f1fb2c747b687b7b8d67640d5f1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:11.819ex; height:2.509ex;" alt="{\displaystyle j=1,\dots ,n}" loading="lazy"></span>.
</p><p>As we have discussed before, most of the entries of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L}</annotation>
</semantics>
</math></span><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M}</annotation>
</semantics>
</math></span><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> are zero because the basis functions <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v_{k}}</annotation>
</semantics>
</math></span><img src="./d142b4083872eb72f81c1e20fd2c91d02b4a9838.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.009ex;" alt="{\displaystyle v_{k}}" loading="lazy"></span> have small support. So we now have to solve a linear system in the unknown <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">u</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} }</annotation>
</semantics>
</math></span><img src="./261e20fe101de02a771021d9d4466c0ad3e352d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }" loading="lazy"></span> where most of the entries of the matrix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L}</annotation>
</semantics>
</math></span><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span>, which we need to invert, are zero.
</p><p>Such matrices are known as <a href="Sparse_matrix" title="Sparse matrix">sparse matrices</a>, and there are efficient solvers for such problems (much more efficient than actually inverting the matrix.) In addition, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L}</annotation>
</semantics>
</math></span><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span> is symmetric and positive definite, so a technique such as the <a href="Conjugate_gradient_method" title="Conjugate gradient method">conjugate gradient method</a> is favored. For problems that are not too large, sparse <a href="LU_decomposition" title="LU decomposition">LU decompositions</a> and <a href="Cholesky_decomposition" title="Cholesky decomposition">Cholesky decompositions</a> still work well. For instance, <a href="MATLAB" title="MATLAB">MATLAB</a>'s backslash operator (which uses sparse LU, sparse Cholesky, and other factorization methods) can be sufficient for meshes with a hundred thousand vertices.
</p><p>The matrix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L}</annotation>
</semantics>
</math></span><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span> is usually referred to as the <a href="Stiffness_matrix" title="Stiffness matrix">stiffness matrix</a>, while the matrix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M}</annotation>
</semantics>
</math></span><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> is dubbed the <a href="Mass_matrix" title="Mass matrix">mass matrix</a>.
</p>
<div class="mw-heading mw-heading3"><h3 id="General_form_of_the_finite_element_method">General form of the finite element method</h3></div>
<p>In general, the finite element method is characterized by the following process.
</p>
<ul><li>One chooses a grid for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>
</semantics>
</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span>. In the preceding treatment, the grid consisted of triangles, but one can also use squares or curvilinear polygons.</li>
<li>Then, one chooses basis functions. We used piecewise linear basis functions in our discussion, but it is common to use piecewise polynomial basis functions.</li></ul>
<p>Separate consideration is the smoothness of the basis functions. For second-order <a href="Elliptic_boundary_value_problem" title="Elliptic boundary value problem">elliptic boundary value problems</a>, piecewise polynomial basis function that is merely continuous suffice (i.e., the derivatives are discontinuous.) For higher-order partial differential equations, one must use smoother basis functions. For instance, for a fourth-order problem such as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{xxxx}+u_{yyyy}=f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>u</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
<mi>x</mi>
<mi>x</mi>
<mi>x</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>u</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
<mi>y</mi>
<mi>y</mi>
<mi>y</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u_{xxxx}+u_{yyyy}=f}</annotation>
</semantics>
</math></span><img src="./09d1831f74bd8583801790de444e619ec2f17611.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.37ex; height:2.843ex;" alt="{\displaystyle u_{xxxx}+u_{yyyy}=f}" loading="lazy"></span>, one may use piecewise quadratic basis functions that are <a href="Smooth_function" class="mw-redirect" title="Smooth function"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>C</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C^{1}}</annotation>
</semantics>
</math></span><img src="./bd24bae0d7570018e828e19851902c09c618af91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.676ex;" alt="{\displaystyle C^{1}}" loading="lazy"></span></a>.
</p><p>Another consideration is the relation of the finite-dimensional space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
</semantics>
</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> to its infinite-dimensional counterpart in the examples above <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}^{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H_{0}^{1}}</annotation>
</semantics>
</math></span><img src="./fd9c9e1684f33b9116efe894347370f5bfc0c8e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.158ex; height:3.176ex;" alt="{\displaystyle H_{0}^{1}}" loading="lazy"></span>. A conforming element method is one in which space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
</semantics>
</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> is a subspace of the element space for the continuous problem. The example above is such a method. If this condition is not satisfied, we obtain a nonconforming element method, an example of which is the space of piecewise linear functions over the mesh, which are continuous at each edge midpoint. Since these functions are generally discontinuous along the edges, this finite-dimensional space is not a subspace of the original <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}^{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H_{0}^{1}}</annotation>
</semantics>
</math></span><img src="./fd9c9e1684f33b9116efe894347370f5bfc0c8e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.158ex; height:3.176ex;" alt="{\displaystyle H_{0}^{1}}" loading="lazy"></span>.
</p><p>Typically, one has an algorithm for subdividing a given mesh. If the primary method for increasing precision is to subdivide the mesh, one has an <i>h</i>-method (<i>h</i> is customarily the diameter of the largest element in the mesh.) In this manner, if one shows that the error with a grid <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h}</annotation>
</semantics>
</math></span><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"></span> is bounded above by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ch^{p}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
<msup>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Ch^{p}}</annotation>
</semantics>
</math></span><img src="./08cd8eb70811c4b4b41fd9e43382084ecd068c2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.164ex; height:2.343ex;" alt="{\displaystyle Ch^{p}}" loading="lazy"></span>, for some <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C<\infty }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
<mo>&lt;</mo>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C&lt;\infty }</annotation>
</semantics>
</math></span><img src="./edbfe849f5960e5d8dddf496797342677b0b2fa7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.189ex; height:2.176ex;" alt="{\displaystyle C<\infty }" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p>0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo>&gt;</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p&gt;0}</annotation>
</semantics>
</math></span><img src="./8dffb51e20581d50c3012634fd9f7b059a68c1c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p>0}" loading="lazy"></span>, then one has an order <i>p</i> method. Under specific hypotheses (for instance, if the domain is convex), a piecewise polynomial of order <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d}</annotation>
</semantics>
</math></span><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"></span> method will have an error of order <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=d+1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo>=</mo>
<mi>d</mi>
<mo>+</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p=d+1}</annotation>
</semantics>
</math></span><img src="./94fe11ba16f2cfa26d675787bda9cc3bcf4fc0e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.576ex; height:2.509ex;" alt="{\displaystyle p=d+1}" loading="lazy"></span>.
</p><p>If instead of making <i>h</i> smaller, one increases the degree of the polynomials used in the basis function, one has a <i>p</i>-method. If one combines these two refinement types, one obtains an <i>hp</i>-method (<a href="Hp-FEM" title="Hp-FEM">hp-FEM</a>). In the hp-FEM, the polynomial degrees can vary from element to element. High-order methods with large uniform <i>p</i> are called spectral finite element methods (<a href="Spectral_element_method" title="Spectral element method">SFEM</a>). These are not to be confused with <a href="Spectral_method" title="Spectral method">spectral methods</a>.
</p><p>For vector partial differential equations, the basis functions may take values in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>
</semantics>
</math></span><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Various_types_of_finite_element_methods">Various types of finite element methods</h2></div>
<div class="mw-heading mw-heading3"><h3 id="AEM">AEM</h3></div>
<p>The Applied Element Method or AEM combines features of both FEM and <a href="Discrete_element_method" title="Discrete element method">Discrete element method</a> or (DEM).
</p>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Applied_element_method" title="Applied element method">Applied element method</a></div>
<div class="mw-heading mw-heading3"><h3 id="A-FEM">A-FEM</h3></div>
<p>Yang and Lui introduced the Augmented-Finite Element Method, whose goal was to model the weak and strong discontinuities without needing extra DoFs, as PuM stated.
</p>
<div class="mw-heading mw-heading3"><h3 id="CutFEM">CutFEM</h3></div>
<p>The Cut Finite Element Approach was developed in 2014.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> The approach is "to make the discretization as independent as possible of the geometric description and minimize the complexity of mesh generation, while retaining the accuracy and robustness of a standard finite element method."<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading3"><h3 id="Generalized_finite_element_method">Generalized finite element method</h3></div>
<p>The generalized finite element method (GFEM) uses local spaces consisting of functions, not necessarily polynomials, that reflect the available information on the unknown solution and thus ensure good local approximation. Then a <a href="Partition_of_unity" title="Partition of unity">partition of unity</a> is used to “bond” these spaces together to form the approximating subspace. The effectiveness of GFEM has been shown when applied to problems with domains having complicated boundaries, problems with micro-scales, and problems with boundary layers.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading3"><h3 id="Mixed_finite_element_method">Mixed finite element method</h3></div>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Mixed_finite_element_method" title="Mixed finite element method">Mixed finite element method</a></div>
<p>The mixed finite element method is a type of finite element method in which extra independent variables are introduced as nodal variables during the discretization of a partial differential equation problem.
</p>
<div class="mw-heading mw-heading3"><h3 id="Variable_–_polynomial">Variable – polynomial</h3></div>
<p>The <a href="Hp-FEM" title="Hp-FEM">hp-FEM</a> combines adaptively elements with variable size <i>h</i> and polynomial degree <i>p</i> to achieve exceptionally fast, exponential convergence rates.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading3"><h3 id="hpk-FEM">hpk-FEM</h3></div>
<p>The hpk-FEM combines adaptively elements with variable size <i>h</i>, polynomial degree of the local approximations <i>p</i>, and global differentiability of the local approximations (<i>k</i>-1) to achieve the best convergence rates.
</p>
<div class="mw-heading mw-heading3"><h3 id="XFEM">XFEM</h3></div>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Extended_finite_element_method" title="Extended finite element method">Extended finite element method</a></div>
<p>The <a href="Extended_finite_element_method" title="Extended finite element method">extended finite element method</a> (XFEM) is a numerical technique based on the generalized finite element method (GFEM) and the partition of unity method (PUM). It extends the classical finite element method by enriching the solution space for solutions to differential equations with discontinuous functions. Extended finite element methods enrich the approximation space to naturally reproduce the challenging feature associated with the problem of interest: the discontinuity, singularity, boundary layer, etc. It was shown that for some problems, such an embedding of the problem's feature into the approximation space can significantly improve convergence rates and accuracy. Moreover, treating problems with discontinuities with XFEMs suppresses the need to mesh and re-mesh the discontinuity surfaces, thus alleviating the computational costs and projection errors associated with conventional finite element methods at the cost of restricting the discontinuities to mesh edges.
</p><p>Several research codes implement this technique to various degrees:
</p>
<ol><li>GetFEM++</li>
<li>xfem++</li>
<li>openxfem++</li></ol>
<p>XFEM has also been implemented in codes like Altair Radios, ASTER, Morfeo, and Abaqus. It is increasingly being adopted by other commercial finite element software, with a few plugins and actual core implementations available (ANSYS, SAMCEF, OOFELIE, etc.).
</p>
<div class="mw-heading mw-heading3"><h3 id="Scaled_boundary_finite_element_method_(SBFEM)">Scaled boundary finite element method (SBFEM)</h3></div>
<p>The introduction of the scaled boundary finite element method (SBFEM) came from Song and Wolf (1997).<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> The SBFEM has been one of the most profitable contributions in the area of numerical analysis of fracture mechanics problems. It is a semi-analytical fundamental-solutionless method combining the advantages of finite element formulations and procedures and boundary element discretization. However, unlike the boundary element method, no fundamental differential solution is required.
</p>
<div class="mw-heading mw-heading3"><h3 id="S-FEM">S-FEM</h3></div>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Smoothed_finite_element_method" title="Smoothed finite element method">Smoothed finite element method</a></div>
<p>The S-FEM, Smoothed Finite Element Methods, is a particular class of numerical simulation algorithms for the simulation of physical phenomena. It was developed by combining mesh-free methods with the finite element method.
</p>
<div class="mw-heading mw-heading3"><h3 id="Spectral_element_method">Spectral element method</h3></div>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Spectral_element_method" title="Spectral element method">Spectral element method</a></div><p>Spectral element methods combine the geometric flexibility of finite elements and the acute accuracy of spectral methods. Spectral methods are the approximate solution of weak-form partial equations based on high-order Lagrangian interpolants and used only with certain quadrature rules.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup>
</p><div class="mw-heading mw-heading3"><h3 id="Meshfree_methods">Meshfree methods</h3></div>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Meshfree_methods" title="Meshfree methods">Meshfree methods</a></div>
<div class="mw-heading mw-heading3"><h3 id="Discontinuous_Galerkin_methods">Discontinuous Galerkin methods</h3></div>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Discontinuous_Galerkin_method" title="Discontinuous Galerkin method">Discontinuous Galerkin method</a></div>
<div class="mw-heading mw-heading3"><h3 id="Finite_element_limit_analysis">Finite element limit analysis</h3></div>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Finite_element_limit_analysis" title="Finite element limit analysis">Finite element limit analysis</a></div>
<div class="mw-heading mw-heading3"><h3 id="Stretched_grid_method">Stretched grid method</h3></div>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Stretched_grid_method" title="Stretched grid method">Stretched grid method</a></div>
<div class="mw-heading mw-heading3"><h3 id="Loubignac_iteration">Loubignac iteration</h3></div>
<p><a href="Loubignac_iteration" title="Loubignac iteration">Loubignac iteration</a> is an iterative method in finite element methods.
</p>
<div class="mw-heading mw-heading3"><h3 id="Crystal_plasticity_finite_element_method_(CPFEM)">Crystal plasticity finite element method (CPFEM)</h3></div>
<p>The crystal plasticity finite element method (CPFEM) is an advanced numerical tool developed by Franz Roters. Metals can be regarded as crystal aggregates, which behave anisotropy under deformation, such as abnormal stress and strain localization. CPFEM, based on the slip (shear strain rate), can calculate dislocation, crystal orientation, and other texture information to consider crystal anisotropy during the routine. It has been applied in the numerical study of material deformation, surface roughness, fractures, etc.
</p>
<div class="mw-heading mw-heading3"><h3 id="Virtual_element_method_(VEM)">Virtual element method (VEM)</h3></div>
<p>The virtual element method (VEM), introduced by Beirão da Veiga et al. (2013)<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> as an extension of <a href="Mimesis_(mathematics)" title="Mimesis (mathematics)">mimetic</a> <a href="Finite_difference_method" title="Finite difference method">finite difference</a> (MFD) methods, is a generalization of the standard finite element method for arbitrary element geometries. This allows admission of general polygons (or <a href="Polyhedra" class="mw-redirect" title="Polyhedra">polyhedra</a> in 3D) that are highly irregular and non-convex in shape. The name <i>virtual</i> derives from the fact that knowledge of the local shape function basis is not required and is, in fact, never explicitly calculated.
</p>
<div class="mw-heading mw-heading2"><h2 id="Link_with_the_gradient_discretization_method">Link with the gradient discretization method</h2></div>
<p>Some types of finite element methods (conforming, nonconforming, mixed finite element methods) are particular cases of the <a href="Gradient_discretization_method" class="mw-redirect" title="Gradient discretization method">gradient discretization method</a> (GDM). Hence the convergence properties of the GDM, which are established for a series of problems (linear and nonlinear elliptic problems, linear, nonlinear, and degenerate parabolic problems), hold as well for these particular FEMs.
</p>
<div class="mw-heading mw-heading2"><h2 id="Comparison_to_the_finite_difference_method">Comparison to the finite difference method</h2></div>
<p>The <a href="Finite_difference_method" title="Finite difference method">finite difference method</a> (FDM) is an alternative way of approximating solutions of PDEs. The differences between FEM and FDM are:
</p>
<ul><li>The most attractive feature of the FEM is its ability to handle complicated geometries (and boundaries) with relative ease. While FDM in its basic form is restricted to handle rectangular shapes and simple alterations thereof, the handling of geometries in FEM is theoretically straightforward.<sup id="cite_ref-:0_3-1" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_23-0" class="reference"><a href="#cite_note-:1-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></li>
<li>FDM is not usually used for irregular CAD geometries but more often for rectangular or block-shaped models.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup></li>
<li>FEM generally allows for more flexible mesh adaptivity than FDM.<sup id="cite_ref-:1_23-1" class="reference"><a href="#cite_note-:1-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></li>
<li>The most attractive feature of finite differences is that it is straightforward to implement.<sup id="cite_ref-:1_23-2" class="reference"><a href="#cite_note-:1-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></li>
<li>One could consider the FDM a particular case of the FEM approach in several ways. E.g., first-order FEM is identical to FDM for <a href="Poisson's_equation" title="Poisson's equation">Poisson's equation</a> if the problem is <a href="Discretization" title="Discretization">discretized</a> by a regular rectangular mesh with each rectangle divided into two triangles.</li>
<li>There are reasons to consider the mathematical foundation of the finite element approximation more sound, for instance, because the quality of the approximation between grid points is poor in FDM.</li>
<li>The quality of a FEM approximation is often higher than in the corresponding FDM approach, but this is highly problem-dependent, and several examples to the contrary can be provided.</li></ul>
<p>Generally, FEM is the method of choice in all types of analysis in structural mechanics (i.e., solving for deformation and stresses in solid bodies or dynamics of structures). In contrast, <a href="Computational_fluid_dynamics" title="Computational fluid dynamics">computational fluid dynamics</a> (CFD) tend to use FDM or other methods like <a href="Finite_volume_method" title="Finite volume method">finite volume method</a> (FVM). CFD problems usually require discretization of the problem into a large number of cells/gridpoints (millions and more). Therefore the cost of the solution favors simpler, lower-order approximation within each cell. This is especially true for 'external flow' problems, like airflow around the car, airplane, or weather simulation.
</p>
<div class="mw-heading mw-heading2"><h2 id="Finite_element_and_fast_fourier_transform_(FFT)_methods">Finite element and fast fourier transform (FFT) methods</h2></div>
<p>Another method used for approximating solutions to a partial differential equation is the <a href="Fast_Fourier_transform" title="Fast Fourier transform">Fast Fourier Transform</a> (FFT), where the solution is approximated by a fourier series computed using the FFT. For approximating the mechanical response of materials under stress, FFT is often much faster,<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> but FEM may be more accurate.<sup id="cite_ref-:2_26-0" class="reference"><a href="#cite_note-:2-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> One example of the respective advantages of the two methods is in simulation of <a href="Rolling_(metalworking)" title="Rolling (metalworking)">rolling</a> a sheet of <a href="Aluminium" title="Aluminium">aluminum</a> (an FCC metal), and <a href="Wire_drawing" title="Wire drawing">drawing</a> a wire of <a href="Tungsten" title="Tungsten">tungsten</a> (a BCC metal). This simulation did not have a sophisticated shape update algorithm for the FFT method. In both cases, the FFT method was more than 10 times as fast as FEM, but in the wire drawing simulation, where there were large deformations in <a href="Crystallite" title="Crystallite">grains</a>, the FEM method was much more accurate. In the sheet rolling simulation, the results of the two methods were similar.<sup id="cite_ref-:2_26-1" class="reference"><a href="#cite_note-:2-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> FFT has a larger speed advantage in cases where the boundary conditions are given in the materials <a href="Strain_(mechanics)" title="Strain (mechanics)">strain</a>, and loses some of its efficiency in cases where the <a href="Stress_(mechanics)" title="Stress (mechanics)">stress</a> is used to apply the boundary conditions, as more iterations of the method are needed.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup>
</p><p>The FE and FFT methods can also be combined in a <a href="Voxel" title="Voxel">voxel</a> based method (2) to simulate deformation in materials, where the FE method is used for the macroscale stress and deformation, and the FFT method is used on the microscale to deal with the effects of microscale on the mechanical response.<sup id="cite_ref-:3_28-0" class="reference"><a href="#cite_note-:3-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> Unlike FEM, FFT methods’ similarities to image processing methods means that an actual image of the microstructure from a microscope can be input to the solver to get a more accurate stress response. Using a real image with FFT avoids meshing the microstructure, which would be required if using FEM simulation of the microstructure, and might be difficult. Because fourier approximations are inherently periodic, FFT can only be used in cases of periodic microstructure, but this is common in real materials.<sup id="cite_ref-:3_28-1" class="reference"><a href="#cite_note-:3-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> FFT can also be combined with FEM methods by using fourier components as the variational basis for approximating the fields inside an element, which can take advantage of the speed of FFT based solvers.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Application">Application</h2></div>


<p>Various specializations under the umbrella of the mechanical engineering discipline (such as aeronautical, biomechanical, and automotive industries) commonly use integrated FEM in the design and development of their products. Several modern FEM packages include specific components such as thermal, electromagnetic, fluid, and structural working environments. In a structural simulation, FEM helps tremendously in producing stiffness and strength visualizations and minimizing weight, materials, and costs.<sup id="cite_ref-Engineering_Asset_Management_30-0" class="reference"><a href="#cite_note-Engineering_Asset_Management-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup>
</p>

<p>This powerful design tool has significantly improved both the standard of engineering designs and the design process methodology in many industrial applications.<sup id="cite_ref-Hastings_32-0" class="reference"><a href="#cite_note-Hastings-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> The introduction of FEM has substantially decreased the time to take products from concept to the production line.<sup id="cite_ref-Hastings_32-1" class="reference"><a href="#cite_note-Hastings-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> Testing and development have been accelerated primarily through improved initial prototype designs using FEM.<sup id="cite_ref-McLaren-Mercedes_33-0" class="reference"><a href="#cite_note-McLaren-Mercedes-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> In summary, benefits of FEM include increased accuracy, enhanced design and better insight into critical design parameters, virtual prototyping, fewer hardware prototypes, a faster and less expensive design cycle, increased productivity, and increased revenue.<sup id="cite_ref-Hastings_32-2" class="reference"><a href="#cite_note-Hastings-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup>
</p><p>In the 1990s FEM was proposed for use in stochastic modeling for numerically solving probability models<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> and later for reliability assessment.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup>
</p><p>FEM is widely applied for approximating differential equations that describe physical systems. This method is very popular in the community of <a href="Computational_fluid_dynamics" title="Computational fluid dynamics">Computational fluid dynamics</a>, and there are many applications for solving <a href="Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a> with FEM.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> Recently, the application of FEM has been increasing in the researches of computational plasma. Promising numerical results using FEM for <a href="Magnetohydrodynamics" title="Magnetohydrodynamics">Magnetohydrodynamics</a>, <a href="Vlasov_equation" title="Vlasov equation">Vlasov equation</a>, and <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> have been proposed.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
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<ul><li><a href="Applied_element_method" title="Applied element method">Applied element method</a></li>
<li><a href="Boundary_element_method" title="Boundary element method">Boundary element method</a></li>
<li><a href="C%C3%A9a's_lemma" title="Céa's lemma">Céa's lemma</a></li>
<li><a href="Computer_experiment" title="Computer experiment">Computer experiment</a></li>
<li><a href="Direct_stiffness_method" title="Direct stiffness method">Direct stiffness method</a></li>
<li><a href="Discontinuity_layout_optimization" title="Discontinuity layout optimization">Discontinuity layout optimization</a></li>
<li><a href="Discrete_element_method" title="Discrete element method">Discrete element method</a></li>
<li><a href="Finite_difference_method" title="Finite difference method">Finite difference method</a></li>
<li><a href="Finite_element_machine" title="Finite element machine">Finite element machine</a></li>
<li><a href="Finite_element_method_in_structural_mechanics" title="Finite element method in structural mechanics">Finite element method in structural mechanics</a></li>
<li><a href="Finite_volume_method" title="Finite volume method">Finite volume method</a></li>
<li><a href="Finite_volume_method_for_unsteady_flow" title="Finite volume method for unsteady flow">Finite volume method for unsteady flow</a></li>
<li><a href="Infinite_element_method" title="Infinite element method">Infinite element method</a></li>
<li><a href="Interval_finite_element" title="Interval finite element">Interval finite element</a></li>
<li><a href="Isogeometric_analysis" title="Isogeometric analysis">Isogeometric analysis</a></li>
<li><a href="Lattice_Boltzmann_methods" title="Lattice Boltzmann methods">Lattice Boltzmann methods</a></li>
<li><a href="List_of_finite_element_software_packages" title="List of finite element software packages">List of finite element software packages</a></li>
<li><a href="Meshfree_methods" title="Meshfree methods">Meshfree methods</a></li>
<li><a href="Movable_cellular_automaton" title="Movable cellular automaton">Movable cellular automaton</a></li>
<li><a href="Multidisciplinary_design_optimization" title="Multidisciplinary design optimization">Multidisciplinary design optimization</a></li>
<li><a href="Multiphysics" class="mw-redirect" title="Multiphysics">Multiphysics</a></li>
<li><a href="Patch_test_(finite_elements)" title="Patch test (finite elements)">Patch test</a></li>
<li><a href="Rayleigh%E2%80%93Ritz_method" title="Rayleigh–Ritz method">Rayleigh–Ritz method</a></li>
<li><a href="SDC_Verifier" title="SDC Verifier">SDC Verifier</a></li>
<li><a href="Space_mapping" title="Space mapping">Space mapping</a></li>
<li><a href="STRAND7" title="STRAND7">STRAND7</a></li>
<li><a href="Tessellation_(computer_graphics)" title="Tessellation (computer graphics)">Tessellation (computer graphics)</a></li>
<li><a href="Weakened_weak_form" title="Weakened weak form">Weakened weak form</a></li></ul>
</div>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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</style><cite id="CITEREFHoangSchwab2005" class="citation journal cs1">Hoang, Viet Ha; Schwab, Christoph (2005). "High-dimensional finite elements for elliptic problems with multiple scales". <i>Multiscale Modeling &amp; Simulation</i>. <b>3</b> (1). SIAM: <span class="nowrap">168–</span>194. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F030601077">10.1137/030601077</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/20.500.11850%2F147656">20.500.11850/147656</a></span>.</cite></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFDaryl_L._Logan2011" class="citation book cs1">Daryl L. Logan (2011). <i>A first course in the finite element method</i>. Cengage Learning. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>9780495668275</bdi>.</cite></span>
</li>
<li id="cite_note-:0-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFReddy2006" class="citation book cs1">Reddy, J. N. (2006). <i>An Introduction to the Finite Element Method</i> (Third&nbsp;ed.). McGraw-Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>9780071267618</bdi>.</cite></span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite class="citation journal cs1"><span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://doi.org/10.1016/s0168-874x(22)00118-4">"Editorial Board"</a></span>. <i>Finite Elements in Analysis and Design</i>. <b>211</b> 103845. 2022. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fs0168-874x%2822%2900118-4">10.1016/s0168-874x(22)00118-4</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a>&nbsp;<a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0168-874X">0168-874X</a>.</cite></span>
</li>
<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFHuebner2001" class="citation book cs1">Huebner, Kenneth H. (2001). <i>The Finite Element Method for Engineers</i>. Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-0-471-37078-9</bdi>.</cite></span>
</li>
<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFLiuLiPark2022" class="citation journal cs1">Liu, Wing Kam; Li, Shaofan; Park, Harold S. (2022). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11831-022-09740-9">"Eighty Years of the Finite Element Method: Birth, Evolution, and Future"</a>. <i>Archives of Computational Methods in Engineering</i>. <b>29</b> (6): <span class="nowrap">4431–</span>4453. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2107.04960">2107.04960</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11831-022-09740-9">10.1007/s11831-022-09740-9</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a>&nbsp;<a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1134-3060">1134-3060</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a>&nbsp;<a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:235794921">235794921</a>.</cite></span>
</li>
<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFHrennikoff1941" class="citation journal cs1">Hrennikoff, Alexander (1941). "Solution of problems of elasticity by the framework method". <i>Journal of Applied Mechanics</i>. <b>8</b> (4): <span class="nowrap">169–</span>175. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1941JAM.....8A.169H">1941JAM.....8A.169H</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1115%2F1.4009129">10.1115/1.4009129</a>.</cite></span>
</li>
<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFCourant1943" class="citation journal cs1">Courant, R. (1943). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0002-9904-1943-07818-4">"Variational methods for the solution of problems of equilibrium and vibrations"</a>. <i>Bulletin of the American Mathematical Society</i>. <b>49</b> (1): <span class="nowrap">1–</span>23. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0002-9904-1943-07818-4">10.1090/s0002-9904-1943-07818-4</a></span>.</cite></span>
</li>
<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20150930001741/http://emi.nw.ru/INDEX.html?0%2Fresume%2Foganesan.htm">"СПб ЭМИ РАН"</a>. <i>emi.nw.ru</i>. Archived from <a rel="nofollow" class="external text" href="http://emi.nw.ru/INDEX.html?0/resume/oganesan.htm">the original</a> on 30 September 2015<span class="reference-accessdate">. Retrieved <span class="nowrap">17 March</span> 2018</span>.</cite></span>
</li>
<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><cite id="CITEREFHintonIrons1968" class="citation journal cs1">Hinton, Ernest; Irons, Bruce (July 1968). "Least squares smoothing of experimental data using finite elements". <i>Strain</i>. <b>4</b> (3): <span class="nowrap">24–</span>27. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1475-1305.1968.tb01368.x">10.1111/j.1475-1305.1968.tb01368.x</a>.</cite></span>
</li>
<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://nisee.berkeley.edu/elibrary/getpkg?id=SAP4">"SAP-IV Software and Manuals"</a>. NISEE e-Library, The Earthquake Engineering Online Archive. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130309013628/http://nisee.berkeley.edu/elibrary/getpkg?id=SAP4">Archived</a> from the original on 2013-03-09<span class="reference-accessdate">. Retrieved <span class="nowrap">2013-01-24</span></span>.</cite></span>
</li>
<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><cite id="CITEREFGard_PaulsenHåkon_With_AndersenJohn_Petter_CollettIver_Tangen_Stensrud2014" class="citation book cs1">Gard Paulsen; Håkon With Andersen; John Petter Collett; Iver Tangen Stensrud (2014). <i>Building Trust, The history of DNV 1864-2014</i>. Lysaker, Norway: Dinamo Forlag A/S. pp.&nbsp;121, 436. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-82-8071-256-1</bdi>.</cite></span>
</li>
<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><cite id="CITEREFStrangFix1973" class="citation book cs1"><a href="Gilbert_Strang" title="Gilbert Strang">Strang, Gilbert</a>; <a href="George_Fix" title="George Fix">Fix, George</a> (1973). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/analysisoffinite0000stra"><i>An Analysis of The Finite Element Method</i></a></span>. Prentice Hall. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-0-13-032946-2</bdi>.</cite></span>
</li>
<li id="cite_note-ZienkiewiczTaylor2013-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-ZienkiewiczTaylor2013_14-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFOlek_C_ZienkiewiczRobert_L_TaylorJ.Z._Zhu2013" class="citation book cs1">Olek C Zienkiewicz; Robert L Taylor; J.Z. Zhu (31 August 2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7UL5Ls9hOF8C"><i>The Finite Element Method: Its Basis and Fundamentals</i></a>. Butterworth-Heinemann. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-0-08-095135-5</bdi>.</cite></span>
</li>
<li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><cite id="CITEREFBathe2006" class="citation book cs1"><a href="Klaus-J%C3%BCrgen_Bathe" title="Klaus-Jürgen Bathe">Bathe, K.J.</a> (2006). <i>Finite Element Procedures</i>. Cambridge, MA: Klaus-Jürgen Bathe. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-0979004902</bdi>.</cite></span>
</li>
<li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><cite id="CITEREFcelledoni2023" class="citation web cs1">celledoni (2023-02-27). <a rel="nofollow" class="external text" href="https://ecmiindmath.org/2023/02/27/cutfem-discretizing-partial-differential-equations-and-geometry/">"CutFEM: Discretizing Partial Differential Equations and Geometry"</a>. <i>ECMI</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-10-13</span></span>.</cite></span>
</li>
<li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><cite id="CITEREFBurmanClausHansboLarson2015" class="citation journal cs1">Burman, Erik; Claus, Susanne; Hansbo, Peter; Larson, Mats G.; Massing, André (2015-11-16). <a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fnme.4823">"CutFEM: Discretizing geometry and partial differential equations"</a>. <i>International Journal for Numerical Methods in Engineering</i>. <b>104</b> (7): <span class="nowrap">472–</span>501. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015IJNME.104..472B">2015IJNME.104..472B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fnme.4823">10.1002/nme.4823</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a>&nbsp;<a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0029-5981">0029-5981</a>.</cite></span>
</li>
<li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><cite id="CITEREFBabuškaBanerjeeOsborn2004" class="citation journal cs1"><a href="Ivo_Babu%C5%A1ka" title="Ivo Babuška">Babuška, Ivo</a>; Banerjee, Uday; <a href="John_E._Osborn_(mathematician)" title="John E. Osborn (mathematician)">Osborn, John E.</a> (June 2004). "Generalized Finite Element Methods: Main Ideas, Results, and Perspective". <i><a href="International_Journal_of_Computational_Methods" title="International Journal of Computational Methods">International Journal of Computational Methods</a></i>. <b>1</b> (1): <span class="nowrap">67–</span>103. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS0219876204000083">10.1142/S0219876204000083</a>.</cite></span>
</li>
<li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">P. Solin, K. Segeth, I. Dolezel: Higher-Order Finite Element Methods, Chapman &amp; Hall/CRC Press, 2003</span>
</li>
<li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><cite id="CITEREFSongWolf1997" class="citation journal cs1">Song, Chongmin; Wolf, John P. (5 August 1997). "The scaled boundary finite-element method – alias consistent infinitesimal finite-element cell method – for elastodynamics". <i>Computer Methods in Applied Mechanics and Engineering</i>. <b>147</b> (<span class="nowrap">3–</span>4): <span class="nowrap">329–</span>355. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1997CMAME.147..329S">1997CMAME.147..329S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0045-7825%2897%2900021-2">10.1016/S0045-7825(97)00021-2</a>.</cite></span>
</li>
<li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20170810215437/http://lsec.cc.ac.cn/~cjxu/SEM_mem.html">"Spectral Element Methods"</a>. <i>State Key Laboratory of Scientific and Engineering Computing</i>. Archived from <a rel="nofollow" class="external text" href="http://lsec.cc.ac.cn/~cjxu/SEM_mem.html">the original</a> on 2017-08-10<span class="reference-accessdate">. Retrieved <span class="nowrap">2017-07-28</span></span>.</cite></span>
</li>
<li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><cite id="CITEREFBeirão_da_VeigaBrezziCangianiManzini2013" class="citation journal cs1">Beirão da Veiga, L.; Brezzi, F.; Cangiani, A.; Manzini, G.; Marini, L. D.; Russo, A. (2013). "Basic principles of Virtual Element Methods". <i><a href="Mathematical_Models_and_Methods_in_Applied_Sciences" title="Mathematical Models and Methods in Applied Sciences">Mathematical Models and Methods in Applied Sciences</a></i>. <b>23</b> (1): <span class="nowrap">199–</span>214. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS0218202512500492">10.1142/S0218202512500492</a>.</cite></span>
</li>
<li id="cite_note-:1-23"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_23-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_23-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:1_23-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFTopper2005" class="citation journal cs1">Topper, Jürgen (January 2005). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1002/wilm.42820050119">"Option pricing with finite elements"</a></span>. <i>Wilmott</i>. <b>2005</b> (1): <span class="nowrap">84–</span>90. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fwilm.42820050119">10.1002/wilm.42820050119</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a>&nbsp;<a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1540-6962">1540-6962</a>.</cite></span>
</li>
<li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><cite class="citation news cs1"><a rel="nofollow" class="external text" href="http://www.machinedesign.com/fea-and-simulation/what-s-difference-between-fem-fdm-and-fvm">"What's The Difference Between FEM, FDM, and FVM?"</a>. <i>Machine Design</i>. 2016-04-18. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170728024918/http://www.machinedesign.com/fea-and-simulation/what-s-difference-between-fem-fdm-and-fvm">Archived</a> from the original on 2017-07-28<span class="reference-accessdate">. Retrieved <span class="nowrap">2017-07-28</span></span>.</cite></span>
</li>
<li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><cite id="CITEREFMaParvathaneniLomovVasiukov2019" class="citation journal cs1">Ma, X; Parvathaneni, K; Lomov, S; Vasiukov, D; Shakoor, M; Park, C (December 2019). <a rel="nofollow" class="external text" href="https://hal.science/hal-02416258">"Quantitative comparison between fast fourier transform and finite element method for micromechanical modeling of composite"</a>. <i>FiBreMoD Conference</i>.</cite></span>
</li>
<li id="cite_note-:2-26"><span class="mw-cite-backlink">^ <a href="#cite_ref-:2_26-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:2_26-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFPrakashLebensohn2009" class="citation journal cs1">Prakash, A; Lebensohn, R A (2009-09-01). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://iopscience.iop.org/article/10.1088/0965-0393/17/6/064010">"Simulation of micromechanical behavior of polycrystals: finite elements versus fast Fourier transforms"</a></span>. <i>Modelling and Simulation in Materials Science and Engineering</i>. <b>17</b> (6): 064010. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009MSMSE..17f4010P">2009MSMSE..17f4010P</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0965-0393%2F17%2F6%2F064010">10.1088/0965-0393/17/6/064010</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a>&nbsp;<a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0965-0393">0965-0393</a>.</cite></span>
</li>
<li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><cite id="CITEREFCruzadoSeguradoHartlBenzerga2021" class="citation journal cs1">Cruzado, A; Segurado, J; Hartl, D J; Benzerga, A A (2021-06-01). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://iopscience.iop.org/article/10.1088/1361-651X/abe4c7">"A variational fast Fourier transform method for phase-transforming materials"</a></span>. <i>Modelling and Simulation in Materials Science and Engineering</i>. <b>29</b> (4): 045001. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2021MSMSE..29d5001C">2021MSMSE..29d5001C</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1361-651X%2Fabe4c7">10.1088/1361-651X/abe4c7</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a>&nbsp;<a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0965-0393">0965-0393</a>.</cite></span>
</li>
<li id="cite_note-:3-28"><span class="mw-cite-backlink">^ <a href="#cite_ref-:3_28-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:3_28-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFGierdenKochmannWaimannSvendsen2022" class="citation journal cs1">Gierden, Christian; Kochmann, Julian; Waimann, Johanna; Svendsen, Bob; Reese, Stefanie (2022-10-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11831-022-09735-6">"A Review of FE-FFT-Based Two-Scale Methods for Computational Modeling of Microstructure Evolution and Macroscopic Material Behavior"</a>. <i>Archives of Computational Methods in Engineering</i>. <b>29</b> (6): <span class="nowrap">4115–</span>4135. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11831-022-09735-6">10.1007/s11831-022-09735-6</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a>&nbsp;<a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1886-1784">1886-1784</a>.</cite></span>
</li>
<li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><cite id="CITEREFZemande_GeusVondřejcPeerlings2017" class="citation journal cs1">Zeman, J.; de Geus, T. W. J.; Vondřejc, J.; Peerlings, R. H. J.; Geers, M. G. D. (2017-09-07). <a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/10.1002/nme.5481">"A finite element perspective on nonlinear FFT-based micromechanical simulations: A FINITE ELEMENT PERSPECTIVE ON NONLINEAR FFT-BASED SIMULATIONS"</a>. <i>International Journal for Numerical Methods in Engineering</i>. <b>111</b> (10): <span class="nowrap">903–</span>926. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1601.05970">1601.05970</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fnme.5481">10.1002/nme.5481</a>.</cite></span>
</li>
<li id="cite_note-Engineering_Asset_Management-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-Engineering_Asset_Management_30-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFKiritsisEemmanouilidisKoroniosMathew2009" class="citation journal cs1">Kiritsis, D.; Eemmanouilidis, Ch.; Koronios, A.; Mathew, J. (2009). "Engineering Asset Management". <i>Proceedings of the 4th World Congress on Engineering Asset Management (WCEAM)</i>: <span class="nowrap">591–</span>592.</cite></span>
</li>
<li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><cite id="CITEREFNaghibi_BeidokhtiJanssenKhoshgoftarSprengers2016" class="citation journal cs1">Naghibi Beidokhti, Hamid; Janssen, Dennis; Khoshgoftar, Mehdi; Sprengers, Andre; Perdahcioglu, Emin Semih; Boogaard, Ton Van den; Verdonschot, Nico (2016). <a rel="nofollow" class="external text" href="https://ris.utwente.nl/ws/files/6153316/CMBBE2014-Hamid-Submitted.pdf">"A comparison between dynamic implicit and explicit finite element simulations of the native knee joint"</a> <span class="cs1-format">(PDF)</span>. <i>Medical Engineering &amp; Physics</i>. <b>38</b> (10): <span class="nowrap">1123–</span>1130. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.medengphy.2016.06.001">10.1016/j.medengphy.2016.06.001</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a>&nbsp;<a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/27349493">27349493</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180719212657/https://ris.utwente.nl/ws/files/6153316/CMBBE2014-Hamid-Submitted.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2018-07-19<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-09-19</span></span>.</cite></span>
</li>
<li id="cite_note-Hastings-32"><span class="mw-cite-backlink">^ <a href="#cite_ref-Hastings_32-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Hastings_32-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Hastings_32-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Hastings, J. K., Juds, M. A., Brauer, J. R., <i>Accuracy and Economy of Finite Element Magnetic Analysis</i>, 33rd Annual National Relay Conference, April 1985.</span>
</li>
<li id="cite_note-McLaren-Mercedes-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-McLaren-Mercedes_33-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFMcLaren-Mercedes2006" class="citation web cs1">McLaren-Mercedes (2006). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20061030200423/http://www.mclaren.com/features/technical/stress_to_impress.php">"McLaren Mercedes: Feature - Stress to impress"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.mclaren.com/features/technical/stress_to_impress.php">the original</a> on 2006-10-30<span class="reference-accessdate">. Retrieved <span class="nowrap">2006-10-03</span></span>.</cite></span>
</li>
<li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text"><cite id="CITEREFPeng_LongWang_JinliangZhu_Qiding1995" class="citation journal cs1">Peng Long; Wang Jinliang; Zhu Qiding (19 May 1995). "Methods with high accuracy for finite element probability computing". <i>Journal of Computational and Applied Mathematics</i>. <b>59</b> (2): <span class="nowrap">181–</span>189. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0377-0427%2894%2900027-X">10.1016/0377-0427(94)00027-X</a>.</cite></span>
</li>
<li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><cite id="CITEREFHaldarMahadevan2000" class="citation book cs1">Haldar, Achintya; Mahadevan, Sankaran (2000). <i>Reliability Assessment Using Stochastic Finite Element Analysis</i>. John Wiley &amp; Sons. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-0471369615</bdi>.</cite></span>
</li>
<li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><cite id="CITEREFGiraultRaviart1979" class="citation book cs1">Girault, Vivette; Raviart, Pierre-Arnaud (1979). <i>Finite Element Approximation of the Navier-Stokes Equations</i>. Vol.&nbsp;749. Springer Berlin. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-3-540-09557-6</bdi>.</cite></span>
</li>
<li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><cite id="CITEREFCuvelierSegalVan_Steenhoven1986" class="citation book cs1">Cuvelier, Cornelis; Segal, August; Van Steenhoven, Anton A (1986). <i>Finite Element Methods and Navier-Stokes Equations</i>. Vol.&nbsp;22. Springer Science &amp; Business Media. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-1-4020-0309-7</bdi>.</cite></span>
</li>
<li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><cite id="CITEREFGiraultRaviart2012" class="citation book cs1">Girault, Vivette; Raviart, Pierre-Arnaud (2012). <i>Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms</i>. Vol.&nbsp;5. Springer Science &amp; Business Media. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-3-642-64888-5</bdi>.</cite></span>
</li>
<li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><cite id="CITEREFKarakashianMakridakis1999" class="citation journal cs1">Karakashian, Ohannes; Makridakis, Charalambos (1999). "A Space-Time Finite Element Method for the Nonlinear Schrödinger Equation: The Continuous Galerkin Method". <i>SIAM Journal on Numerical Analysis</i>. <b>36</b> (6). SIAM: <span class="nowrap">1779–</span>1807. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2FS0036142997330111">10.1137/S0036142997330111</a>.</cite></span>
</li>
<li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text"><cite id="CITEREFSovinecGlasserGianakonBarnes2004" class="citation journal cs1">Sovinec, Carl R.; Glasser, A.H.; Gianakon, T.A.; Barnes, D.C.; Nebel, R.A.; Kruger, S.E.; Schnack, D.D.; Plimpton, S.J.; Tarditi, A.; Chu, M.S. (2004). "Nonlinear Magnetohydrodynamics Simulation Using High-Order Finite Elements". <i>Journal of Computational Physics</i>. <b>195</b> (1). Elsevier: <span class="nowrap">355–</span>386. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004JCoPh.195..355S">2004JCoPh.195..355S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jcp.2003.10.004">10.1016/j.jcp.2003.10.004</a>.</cite></span>
</li>
</ol></div>
<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>
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<div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Finite_element_modelling" class="extiw external" title="commons:Category:Finite element modelling">Finite element modelling</a></span>.</div></div>
</div>
<ul><li>G. Allaire and A. Craig: <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=HIwSDAAAQBAJ&amp;q=%22finite+element%22">Numerical Analysis and Optimization: An Introduction to Mathematical Modelling and Numerical Simulation</a></i>.</li>
<li>K. J. Bathe: <i>Numerical methods in finite element analysis</i>, Prentice-Hall (1976).</li>
<li>Thomas J.R. Hughes: <i>The Finite Element Method: Linear Static and Dynamic Finite Element Analysis,</i> Prentice-Hall (1987).</li>
<li>J. Chaskalovic: <i>Finite Elements Methods for Engineering Sciences</i>, Springer Verlag, (2008).</li>
<li><a href="Endre_S%C3%BCli" title="Endre Süli">Endre Süli</a>: <a rel="nofollow" class="external text" href="http://people.maths.ox.ac.uk/suli/fem.pdf"><i>Finite Element Methods for Partial Differential Equations</i></a>.</li>
<li>O. C. Zienkiewicz, R. L. Taylor, J. Z. Zhu&nbsp;: <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=YocoaH8lnx8C">The Finite Element Method: Its Basis and Fundamentals</a></i>, Butterworth-Heinemann (2005).</li>
<li>N. Ottosen, H. Petersson: <i>Introduction to the Finite Element Method, </i> Prentice-Hall (1992).</li>
<li>Susanne C. Brenner, L. Ridgway Scott: <i>The Mathematical Theory of Finite Element Methods</i>, Springer-Verlag New York, ISBN 978-0-387-75933-3 (2008).</li>
<li>Zohdi, T. I. (2018) A finite element primer for beginners-extended version including sample tests and projects. Second Edition <a rel="nofollow" class="external free" href="https://link.springer.com/book/10.1007/978-3-319-70428-9">https://link.springer.com/book/10.1007/978-3-319-70428-9</a></li>
<li>Leszek F. Demkowicz: <i>Mathematical Theory of Finite Elements</i>, SIAM, ISBN 978-1-61197-772-1 (2024).</li></ul>
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</style></div><div role="navigation" class="navbox" aria-labelledby="Numerical_methods_for_partial_differential_equations284" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Numerical_methods_for_partial_differential_equations284" style="font-size:114%;margin:0 4em"><a href="Numerical_methods_for_partial_differential_equations" title="Numerical methods for partial differential equations">Numerical methods for partial differential equations</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Finite_difference_method" title="Finite difference method">Finite difference</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Parabolic_partial_differential_equation" title="Parabolic partial differential equation">Parabolic</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="FTCS_scheme" title="FTCS scheme">Forward-time central-space</a> (FTCS)</li>
<li><a href="Crank%E2%80%93Nicolson_method" title="Crank–Nicolson method">Crank–Nicolson</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Hyperbolic_partial_differential_equation" title="Hyperbolic partial differential equation">Hyperbolic</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Lax%E2%80%93Friedrichs_method" title="Lax–Friedrichs method">Lax–Friedrichs</a></li>
<li><a href="Lax%E2%80%93Wendroff_method" title="Lax–Wendroff method">Lax–Wendroff</a></li>
<li><a href="MacCormack_method" title="MacCormack method">MacCormack</a></li>
<li><a href="Upwind_scheme" title="Upwind scheme">Upwind</a></li>
<li><a href="Method_of_characteristics" title="Method of characteristics">Method of characteristics</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Others</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Alternating_direction_implicit_method" class="mw-redirect" title="Alternating direction implicit method">Alternating direction-implicit</a> (ADI)</li>
<li><a href="Finite-difference_frequency-domain_method" title="Finite-difference frequency-domain method">Finite-difference frequency-domain</a> (FDFD)</li>
<li><a href="Finite-difference_time-domain_method" title="Finite-difference time-domain method">Finite-difference time-domain</a> (FDTD)</li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Finite_volume_method" title="Finite volume method">Finite volume</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Godunov's_scheme" title="Godunov's scheme">Godunov</a></li>
<li><a href="High-resolution_scheme" title="High-resolution scheme">High-resolution</a></li>
<li><a href="MUSCL_scheme" title="MUSCL scheme">Monotonic upstream-centered</a> (MUSCL)</li>
<li><a href="AUSM" class="mw-redirect" title="AUSM">Advection upstream-splitting</a> (AUSM)</li>
<li><a href="Riemann_solver" title="Riemann solver">Riemann solver</a></li>
<li><a href="ENO_methods" title="ENO methods">Essentially non-oscillatory</a> (ENO)</li>
<li><a href="WENO_methods" title="WENO methods">Weighted essentially non-oscillatory</a> (WENO)</li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Hp-FEM" title="Hp-FEM">hp-FEM</a></li>
<li><a href="Extended_finite_element_method" title="Extended finite element method">Extended</a> (XFEM)</li>
<li><a href="Discontinuous_Galerkin_method" title="Discontinuous Galerkin method">Discontinuous Galerkin</a> (DG)</li>
<li><a href="Spectral_element_method" title="Spectral element method">Spectral element</a> (SEM)</li>
<li><a href="Mortar_methods" title="Mortar methods">Mortar</a></li>
<li><a href="Gradient_discretisation_method" title="Gradient discretisation method">Gradient discretisation</a> (GDM)</li>
<li><a href="Loubignac_iteration" title="Loubignac iteration">Loubignac iteration</a></li>
<li><a href="Smoothed_finite_element_method" title="Smoothed finite element method">Smoothed</a> (S-FEM)</li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Meshfree_methods" title="Meshfree methods">Meshless/Meshfree</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Smoothed-particle_hydrodynamics" title="Smoothed-particle hydrodynamics">Smoothed-particle hydrodynamics</a> (SPH)</li>
<li><a href="Peridynamics" title="Peridynamics">Peridynamics</a> (PD)</li>
<li><a href="Moving_particle_semi-implicit_method" title="Moving particle semi-implicit method">Moving particle semi-implicit method</a> (MPS)</li>
<li><a href="Material_point_method" title="Material point method">Material point method</a> (MPM)</li>
<li><a href="Particle-in-cell" title="Particle-in-cell">Particle-in-cell</a> (PIC)</li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Domain_decomposition_methods" title="Domain decomposition methods">Domain decomposition</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Schur_complement_method" title="Schur complement method">Schur complement</a></li>
<li><a href="Fictitious_domain_method" title="Fictitious domain method">Fictitious domain</a></li>
<li><a href="Schwarz_alternating_method" title="Schwarz alternating method">Schwarz alternating</a>
<ul><li><a href="Additive_Schwarz_method" title="Additive Schwarz method">additive</a></li>
<li><a href="Abstract_additive_Schwarz_method" title="Abstract additive Schwarz method">abstract additive</a></li></ul></li>
<li><a href="Neumann%E2%80%93Dirichlet_method" title="Neumann–Dirichlet method">Neumann–Dirichlet</a></li>
<li><a href="Neumann%E2%80%93Neumann_methods" title="Neumann–Neumann methods">Neumann–Neumann</a></li>
<li><a href="Poincar%C3%A9%E2%80%93Steklov_operator" title="Poincaré–Steklov operator">Poincaré–Steklov operator</a></li>
<li><a href="Balancing_domain_decomposition_method" title="Balancing domain decomposition method">Balancing</a> (BDD)</li>
<li><a href="BDDC" title="BDDC">Balancing by constraints</a> (BDDC)</li>
<li><a href="FETI" title="FETI">Tearing and interconnect</a> (FETI)</li>
<li><a href="FETI-DP" title="FETI-DP">FETI-DP</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Others</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Spectral_method" title="Spectral method">Spectral</a></li>
<li><a href="Pseudo-spectral_method" title="Pseudo-spectral method">Pseudospectral</a> (DVR)</li>
<li><a href="Method_of_lines" title="Method of lines">Method of lines</a></li>
<li><a href="Multigrid_method" title="Multigrid method">Multigrid</a></li>
<li><a href="Collocation_method" title="Collocation method">Collocation</a></li>
<li><a href="Level-set_method" title="Level-set method">Level-set</a></li>
<li><a href="Boundary_element_method" title="Boundary element method">Boundary element</a>
<ul><li><a href="Method_of_moments_(electromagnetics)" title="Method of moments (electromagnetics)">Method of moments</a></li></ul></li>
<li><a href="Immersed_boundary_method" title="Immersed boundary method">Immersed boundary</a></li>
<li><a href="Analytic_element_method" title="Analytic element method">Analytic element</a></li>
<li><a href="Isogeometric_analysis" title="Isogeometric analysis">Isogeometric analysis</a></li>
<li><a href="Infinite_difference_method" title="Infinite difference method">Infinite difference method</a></li>
<li><a href="Infinite_element_method" title="Infinite element method">Infinite element method</a></li>
<li><a href="Galerkin_method" title="Galerkin method">Galerkin method</a>
<ul><li><a href="Petrov%E2%80%93Galerkin_method" title="Petrov–Galerkin method">Petrov–Galerkin method</a></li></ul></li>
<li><a href="Validated_numerics" title="Validated numerics">Validated numerics</a></li>
<li><a href="Computer-assisted_proof" title="Computer-assisted proof">Computer-assisted proof</a></li>
<li><a href="Integrable_algorithm" title="Integrable algorithm">Integrable algorithm</a></li>
<li><a href="Method_of_fundamental_solutions" title="Method of fundamental solutions">Method of fundamental solutions</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Numerical_methods_for_ordinary_differential_equations" title="Numerical methods for ordinary differential equations">Numerical methods for ordinary differential equations</a></li>
<li><a href="Numerical_integration" title="Numerical integration">Numerical integration</a></li></ul>
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