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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Collocation method</span></span>

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<p>In mathematics, a <b>collocation method</b> is a method for the <a href="Numerical_analysis" title="Numerical analysis">numerical</a> solution of <a href="Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equations</a>, <a href="Partial_differential_equation" title="Partial differential equation">partial differential equations</a> and <a href="Integral_equation" title="Integral equation">integral equations</a>. The idea is to choose a finite-dimensional space of candidate solutions (usually <a href="Polynomial" title="Polynomial">polynomials</a> up to a certain degree) and a number of points in the domain (called <i>collocation points</i>), and to select that solution which satisfies the given equation at the collocation points.

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<div class="mw-heading mw-heading2"><h2 id="Ordinary_differential_equations">Ordinary differential equations</h2></div>

<p>Suppose that the <a href="Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equation</a>

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<annotation encoding="application/x-tex">{\displaystyle y'(t)=f(t,y(t)),\quad y(t_{0})=y_{0},}</annotation>

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</math></span><img src="./16d7dc94cd047fc8f2ea0d20976dbf2e33f30eac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.512ex; height:3.009ex;" alt="{\displaystyle y'(t)=f(t,y(t)),\quad y(t_{0})=y_{0},}" loading="lazy"></span></dd></dl>

<p>is to be solved over 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 [t_{0},t_{0}+h]}">

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<annotation encoding="application/x-tex">{\displaystyle [t_{0},t_{0}+h]}</annotation>

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</math></span><img src="./67dfc6adc51170b9c42340d31fdcb4e095250c18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.295ex; height:2.843ex;" alt="{\displaystyle [t_{0},t_{0}+h]}" 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 c_{k}}">

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<annotation encoding="application/x-tex">{\displaystyle c_{k}}</annotation>

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</math></span><img src="./3d2f8052630e67b00d04e3487e1d68ed7070470b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.096ex; height:2.009ex;" alt="{\displaystyle c_{k}}" loading="lazy"></span> from 0 ≤ <i>c</i><sub>1</sub>< <i>c</i><sub>2</sub>< ... < <i>c</i><sub><i>n</i></sub> ≤ 1.

</p><p>The corresponding (polynomial) collocation method approximates the solution <i>y</i> by the polynomial <i>p</i> of degree <i>n</i> which satisfies the initial condition <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(t_{0})=y_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle p(t_{0})=y_{0}}</annotation>

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</math></span><img src="./7ae777564830ce254a2afe7c6f7aa4d4a0b0c813.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:10.254ex; height:2.843ex;" alt="{\displaystyle p(t_{0})=y_{0}}" loading="lazy"></span>, and the differential equation <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'(t_{k})=f(t_{k},p(t_{k}))}">

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<annotation encoding="application/x-tex">{\displaystyle p'(t_{k})=f(t_{k},p(t_{k}))}</annotation>

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</math></span><img src="./08d9e3016b2abe8464e68f8264f844d319d7abbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:19.737ex; height:3.009ex;" alt="{\displaystyle p'(t_{k})=f(t_{k},p(t_{k}))}" loading="lazy"></span>

at all <i>collocation points</i> <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 t_{k}=t_{0}+c_{k}h}">

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<annotation encoding="application/x-tex">{\displaystyle t_{k}=t_{0}+c_{k}h}</annotation>

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</math></span><img src="./2321534852ad99197bd27b94b3a4519312caeea0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.196ex; height:2.509ex;" alt="{\displaystyle t_{k}=t_{0}+c_{k}h}" 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 k=1,\ldots ,n}">

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<annotation encoding="application/x-tex">{\displaystyle k=1,\ldots ,n}</annotation>

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</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>. This gives <i>n</i> + 1 conditions, which matches the <i>n</i> + 1 parameters needed to specify a polynomial of degree <i>n</i>.

</p><p>All these collocation methods are in fact implicit <a href="Runge%E2%80%93Kutta_methods" title="Runge–Kutta methods">Runge–Kutta methods</a>. The coefficients <i>c</i><sub><i>k</i></sub> in the Butcher tableau of a Runge–Kutta method are the collocation points. However, not all implicit Runge–Kutta methods are collocation methods.

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<div class="mw-heading mw-heading3"><h3 id="Example:_The_trapezoidal_rule">Example: The trapezoidal rule</h3></div>

<p>Pick, as an example, the two collocation points <i>c</i><sub>1</sub> = 0 and <i>c</i><sub>2</sub> = 1 (so <i>n</i> = 2). The collocation conditions are

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<annotation encoding="application/x-tex">{\displaystyle p(t_{0})=y_{0},\,}</annotation>

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</math></span><img src="./856aee4a12855be6f1717345dd71d5e85e0db3e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:11.288ex; height:2.843ex;" alt="{\displaystyle p(t_{0})=y_{0},\,}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle p'(t_{0})=f(t_{0},p(t_{0})),\,}</annotation>

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</math></span><img src="./f5ee02a8324317dd055d3f35d9671faeb5c0cbae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:20.667ex; height:3.009ex;" alt="{\displaystyle p'(t_{0})=f(t_{0},p(t_{0})),\,}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle p'(t_{0}+h)=f(t_{0}+h,p(t_{0}+h)).\,}</annotation>

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</math></span><img src="./d9f8be76eb3762b9a65182779c40fb8aa4892406.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:33.205ex; height:3.009ex;" alt="{\displaystyle p'(t_{0}+h)=f(t_{0}+h,p(t_{0}+h)).\,}" loading="lazy"></span></dd></dl>

<p>There are three conditions, so <i>p</i> should be a polynomial of degree 2. Write <i>p</i> in the form

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<annotation encoding="application/x-tex">{\displaystyle p(t)=\alpha (t-t_{0})^{2}+\beta (t-t_{0})+\gamma \,}</annotation>

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</math></span><img src="./2df388b1a8e71c861e3e6b1c7c9db7fa117b2e97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:32.977ex; height:3.176ex;" alt="{\displaystyle p(t)=\alpha (t-t_{0})^{2}+\beta (t-t_{0})+\gamma \,}" loading="lazy"></span></dd></dl>

<p>to simplify the computations. Then the collocation conditions can be solved to give the coefficients

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\alpha &={\frac {1}{2h}}{\Big (}f(t_{0}+h,p(t_{0}+h))-f(t_{0},p(t_{0})){\Big )},\\\beta &=f(t_{0},p(t_{0})),\\\gamma &=y_{0}.\end{aligned}}}</annotation>

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</math></span><img src="./3d4ffaa970f5eea5a73379ab090bf7ea49315caa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.921ex; margin-bottom: -0.25ex; width:45.073ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}\alpha &={\frac {1}{2h}}{\Big (}f(t_{0}+h,p(t_{0}+h))-f(t_{0},p(t_{0})){\Big )},\\\beta &=f(t_{0},p(t_{0})),\\\gamma &=y_{0}.\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>The collocation method is now given (implicitly) by

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<annotation encoding="application/x-tex">{\displaystyle y_{1}=p(t_{0}+h)=y_{0}+{\frac {1}{2}}h{\Big (}f(t_{0}+h,y_{1})+f(t_{0},y_{0}){\Big )},\,}</annotation>

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</math></span><img src="./28b55a4b5fb1a70e54e57b79852182f5c2ea3b50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:53.061ex; height:5.176ex;" alt="{\displaystyle y_{1}=p(t_{0}+h)=y_{0}+{\frac {1}{2}}h{\Big (}f(t_{0}+h,y_{1})+f(t_{0},y_{0}){\Big )},\,}" loading="lazy"></span></dd></dl>

<p>where <i>y</i><sub>1</sub> = <i>p</i>(<i>t</i><sub>0</sub> + <i>h</i>) is the approximate solution at <i>t</i> = <i>t</i><sub>1</sub> = <i>t</i><sub>0</sub> + <i>h</i>.

</p><p>This method is known as the "<a href="Trapezoidal_rule_(differential_equations)" title="Trapezoidal rule (differential equations)">trapezoidal rule</a>" for differential equations. Indeed, this method can also be derived by rewriting the differential equation as

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<annotation encoding="application/x-tex">{\displaystyle y(t)=y(t_{0})+\int _{t_{0}}^{t}f(\tau ,y(\tau ))\,{\textrm {d}}\tau ,\,}</annotation>

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</math></span><img src="./ded285da945408d355b58163ce0c3d2b6acfa2d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.685ex; height:6.509ex;" alt="{\displaystyle y(t)=y(t_{0})+\int _{t_{0}}^{t}f(\tau ,y(\tau ))\,{\textrm {d}}\tau ,\,}" loading="lazy"></span></dd></dl>

<p>and approximating the integral on the right-hand side by the <a href="Trapezoidal_rule" title="Trapezoidal rule">trapezoidal rule</a> for integrals.

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<div class="mw-heading mw-heading3"><h3 id="Other_examples">Other examples</h3></div>

<p>The <a href="Gauss%E2%80%93Legendre_method" title="Gauss–Legendre method">Gauss–Legendre methods</a> use the points of <a href="Gauss%E2%80%93Legendre_quadrature" title="Gauss–Legendre quadrature">Gauss–Legendre quadrature</a> as collocation points. The Gauss–Legendre method based on <i>s</i> points has order 2<i>s</i>.<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> All Gauss–Legendre methods are <a href="A-stability" class="mw-redirect" title="A-stability">A-stable</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>In fact, one can show that the order of a collocation method corresponds to the order of the quadrature rule that one would get using the collocation points as weights.

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<div class="mw-heading mw-heading2"><h2 id="Orthogonal_collocation_method">Orthogonal collocation method</h2></div>

<p>In direct collocation method, we are essentially performing variational calculus with the finite-dimensional subspace of piecewise linear functions (as in trapezoidal rule), or cubic functions, or other piecewise polynomial functions. In orthogonal collocation method, we instead use the finite-dimensional subspace spanned by the first N vectors in some <a href="Orthogonal_polynomials" title="Orthogonal polynomials">orthogonal polynomial</a> basis, such as the <a href="Legendre_polynomials" title="Legendre polynomials">Legendre polynomials</a>.

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<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFAscherPetzold1998">Ascher & Petzold 1998</a>; <a href="#CITEREFIserles1996">Iserles 1996</a>, pp. 43–44</span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFIserles1996">Iserles 1996</a>, pp. 47</span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="#CITEREFIserles1996">Iserles 1996</a>, pp. 63</span>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFAscherPetzold1998" class="citation cs2">Ascher, Uri M.; <a href="Linda_Petzold" title="Linda Petzold">Petzold, Linda R.</a> (1998), <i>Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations</i>, Philadelphia: <a href="Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">Society for Industrial and Applied Mathematics</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-89871-412-8</bdi></cite>.</li>

<li><cite id="CITEREFHairerNørsettWanner1993" class="citation cs2">Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), <i>Solving ordinary differential equations I: Nonstiff problems</i>, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-56670-0</bdi></cite>.</li>

<li><cite id="CITEREFIserles1996" class="citation cs2"><a href="Arieh_Iserles" title="Arieh Iserles">Iserles, Arieh</a> (1996), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7Zofw3SFTWIC&q=%22Collocation+method%22"><i>A First Course in the Numerical Analysis of Differential Equations</i></a>, <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <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/1996fcna.book.....I">1996fcna.book.....I</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-55655-2</bdi></cite>.</li>

<li><cite id="CITEREFWangChenWu2009" class="citation cs2">Wang, Yingwei; Chen, Suqin; Wu, Xionghua (2009), "A rational spectral collocation method for solving a class of parameterized singular perturbation problems", <i>Journal of Computational and Applied Mathematics</i>, <b>233</b> (10): <span class="nowrap">2652–</span>2660, <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/10.1016%2Fj.cam.2009.11.011">10.1016/j.cam.2009.11.011</a></span></cite>.</li></ul>

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</style><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%"><a href="Finite_element_method" title="Finite element method">Finite element</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<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>

</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="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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