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<title>Discrete Poisson equation</title>

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Discrete Poisson equation

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In <a href="Mathematics" title="Mathematics">mathematics</a>, the discrete Poisson equation is the <a href="Finite_difference" title="Finite difference">finite difference</a> analog of the <a href="Poisson_equation" class="mw-redirect" title="Poisson equation">Poisson equation</a>. In it, the <a href="Discrete_Laplace_operator" title="Discrete Laplace operator">discrete Laplace operator</a> takes the place of the <a href="Laplace_operator" title="Laplace operator">Laplace operator</a>. The discrete Poisson equation is frequently used in <a href="Numerical_analysis" title="Numerical analysis">numerical analysis</a> as a stand-in for the continuous Poisson equation, although it is also studied in its own right as a topic in <a href="Discrete_mathematics" title="Discrete mathematics">discrete mathematics</a>.

<div class="mw-heading mw-heading2"><h2 id="On_a_two-dimensional_rectangular_grid">On a two-dimensional rectangular grid</h2></div>

Using the <a href="Finite_difference" title="Finite difference">finite difference</a> numerical method to discretize

the 2-dimensional Poisson equation (assuming a uniform spatial discretization, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x=\Delta y}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta x=\Delta y}</annotation>

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</math><img src="./10a03949208bf49c41db3e9b08d896cc200c26d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.455ex; height:2.509ex;" alt="{\displaystyle \Delta x=\Delta y}" loading="lazy">) on an m × n grid gives the following formula:<a href="#cite_note-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle ({\nabla }^{2}u)_{ij}={\frac {1}{\Delta x^{2}}}(u_{i+1,j}+u_{i-1,j}+u_{i,j+1}+u_{i,j-1}-4u_{ij})=g_{ij}}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\leq i\leq m-1}">

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<annotation encoding="application/x-tex">{\displaystyle 2\leq i\leq m-1}</annotation>

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</math><img src="./1ea08b23fd131dfcf1dce61ea81579bff7a4c044.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.205ex; height:2.343ex;" alt="{\displaystyle 2\leq i\leq m-1}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\leq j\leq n-1}">

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<annotation encoding="application/x-tex">{\displaystyle 2\leq j\leq n-1}</annotation>

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</math><img src="./5f5efcd9320e4036323d750c9708814646ee3aa4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.715ex; height:2.509ex;" alt="{\displaystyle 2\leq j\leq n-1}" loading="lazy">. The preferred arrangement of the solution vector is to use <a href="Enumeration" title="Enumeration">natural ordering</a> which, prior to removing boundary elements, would look like:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} ={\begin{bmatrix}u_{11},u_{21},\ldots ,u_{m1},u_{12},u_{22},\ldots ,u_{m2},\ldots ,u_{mn}\end{bmatrix}}^{\mathsf {T}}}</annotation>

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This will result in an mn × mn linear system:

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<annotation encoding="application/x-tex">{\displaystyle A\mathbf {u} =\mathbf {b} }</annotation>

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where

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<annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}~D&-I&~0&~0&~0&\cdots &~0\\-I&~D&-I&~0&~0&\cdots &~0\\~0&-I&~D&-I&~0&\cdots &~0\\\vdots &\ddots &\ddots &\ddots &\ddots &\ddots &\vdots \\~0&\cdots &~0&-I&~D&-I&~0\\~0&\cdots &\cdots &~0&-I&~D&-I\\~0&\cdots &\cdots &\cdots &~0&-I&~D\end{bmatrix}},}</annotation>

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

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</math><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> is the m × m <a href="Identity_matrix" title="Identity matrix">identity matrix</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

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

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</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy">, also m × m, is given by:<a href="#cite_note-2">[2]</a>

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<annotation encoding="application/x-tex">{\displaystyle D={\begin{bmatrix}~4&-1&~0&~0&~0&\cdots &~0\\-1&~4&-1&~0&~0&\cdots &~0\\~0&-1&~4&-1&~0&\cdots &~0\\\vdots &\ddots &\ddots &\ddots &\ddots &\ddots &\vdots \\~0&\cdots &~0&-1&~4&-1&~0\\~0&\cdots &\cdots &~0&-1&~4&-1\\~0&\cdots &\cdots &\cdots &~0&-1&~4\end{bmatrix}},}</annotation>

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and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {b} }</annotation>

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</math><img src="./13ebf4628a1adf07133a6009e4a78bdd990c6eb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle \mathbf {b} }" loading="lazy"> is defined by

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {b} =-\Delta x^{2}{\begin{bmatrix}g_{11},g_{21},\ldots ,g_{m1},g_{12},g_{22},\ldots ,g_{m2},\ldots ,g_{mn}\end{bmatrix}}^{\mathsf {T}}.}</annotation>

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For each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{ij}}">

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<annotation encoding="application/x-tex">{\displaystyle u_{ij}}</annotation>

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</math><img src="./6dda9331f162d678a64d0c1ffabfe67c1ca570b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.807ex; height:2.343ex;" alt="{\displaystyle u_{ij}}" loading="lazy"> equation, the columns of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

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

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</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> correspond to a block of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> components in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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

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<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}u_{1j},&u_{2j},&\ldots ,&u_{i-1,j},&u_{ij},&u_{i+1,j},&\ldots ,&u_{mj}\end{bmatrix}}^{\mathsf {T}}}</annotation>

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while the columns of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

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

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</math><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> to the left and right of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

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

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</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> each correspond to other blocks of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> components within <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}u_{1,j-1},&u_{2,j-1},&\ldots ,&u_{i-1,j-1},&u_{i,j-1},&u_{i+1,j-1},&\ldots ,&u_{m,j-1}\end{bmatrix}}^{\mathsf {T}}}</annotation>

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and

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<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}u_{1,j+1},&u_{2,j+1},&\ldots ,&u_{i-1,j+1},&u_{i,j+1},&u_{i+1,j+1},&\ldots ,&u_{m,j+1}\end{bmatrix}}^{\mathsf {T}}}</annotation>

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respectively.

From the above, it can be inferred that there are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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</math><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"> block columns of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">. It is important to note that prescribed values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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

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</math><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"> (usually lying on the boundary) would have their corresponding elements removed from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

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

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</math><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

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

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy">. For the common case that all the nodes on the boundary are set, we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\leq i\leq m-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\leq i\leq m-1}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\leq j\leq n-1}</annotation>

</semantics>

</math><img src="./5f5efcd9320e4036323d750c9708814646ee3aa4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.715ex; height:2.509ex;" alt="{\displaystyle 2\leq j\leq n-1}" loading="lazy">, and the system would have the dimensions (m − 2)(n − 2) × (m− 2)(n − 2), where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

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

</semantics>

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

For a 3×3 ( <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=3}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m=3}</annotation>

</semantics>

</math><img src="./918bb386a7ca6891255b62ef91ccc022883f3809.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=3}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=3}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=3}</annotation>

</semantics>

</math><img src="./1c5a5a42ced00df920fad4ab2d4acdb960a4105b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=3}" loading="lazy"> ) grid with all the boundary nodes prescribed, the system would look like:

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msup>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}U\end{bmatrix}}={\begin{bmatrix}u_{22},u_{32},u_{42},u_{23},u_{33},u_{43},u_{24},u_{34},u_{44}\end{bmatrix}}^{\mathsf {T}}}</annotation>

</semantics>

</math>

with

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

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<mtd>

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<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A=\left[{\begin{array}{ccc|ccc|ccc}~4&-1&~0&-1&~0&~0&~0&~0&~0\\-1&~4&-1&~0&-1&~0&~0&~0&~0\\~0&-1&~4&~0&~0&-1&~0&~0&~0\\\hline -1&~0&~0&~4&-1&~0&-1&~0&~0\\~0&-1&~0&-1&~4&-1&~0&-1&~0\\~0&~0&-1&~0&-1&~4&~0&~0&-1\\\hline ~0&~0&~0&-1&~0&~0&~4&-1&~0\\~0&~0&~0&~0&-1&~0&-1&~4&-1\\~0&~0&~0&~0&~0&-1&~0&-1&~4\end{array}}\right]}</annotation>

</semantics>

</math>

and

</mrow>

<mrow>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {b} =\left[{\begin{array}{l}-\Delta x^{2}g_{22}+u_{12}+u_{21}\\-\Delta x^{2}g_{32}+u_{31}~~~~~~~~\\-\Delta x^{2}g_{42}+u_{52}+u_{41}\\-\Delta x^{2}g_{23}+u_{13}~~~~~~~~\\-\Delta x^{2}g_{33}~~~~~~~~~~~~~~~~\\-\Delta x^{2}g_{43}+u_{53}~~~~~~~~\\-\Delta x^{2}g_{24}+u_{14}+u_{25}\\-\Delta x^{2}g_{34}+u_{35}~~~~~~~~\\-\Delta x^{2}g_{44}+u_{54}+u_{45}\end{array}}\right].}</annotation>

</semantics>

</math></dd></dl>

As can be seen, the boundary <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

</mrow>

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

</semantics>

</math><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">'s are brought to the right-hand-side of the equation.<a href="#cite_note-3">[3]</a> The entire system is 9 × 9 while <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

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

</semantics>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D={\begin{bmatrix}~4&-1&~0\\-1&~4&-1\\~0&-1&~4\\\end{bmatrix}}}</annotation>

</semantics>

</math>

and

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -I={\begin{bmatrix}-1&~0&~0\\~0&-1&~0\\~0&~0&-1\end{bmatrix}}.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading2"><h2 id="Methods_of_solution">Methods of solution</h2></div>

Because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}A\end{bmatrix}}}">

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}A\end{bmatrix}}}</annotation>

</semantics>

</math><img src="./66ec302063e76394a96c4472c061f3fb0e32abd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.788ex; height:2.843ex;" alt="{\displaystyle {\begin{bmatrix}A\end{bmatrix}}}" loading="lazy"> is block tridiagonal and sparse, many methods of solution

have been developed to optimally solve this linear system for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}U\end{bmatrix}}}">

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}U\end{bmatrix}}}</annotation>

</semantics>

</math><img src="./c54dd5d4bc07312e67d72fa3c9320dc91c9461ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.828ex; height:2.843ex;" alt="{\displaystyle {\begin{bmatrix}U\end{bmatrix}}}" loading="lazy">.

Among the methods are a generalized <a href="Thomas_algorithm" class="mw-redirect" title="Thomas algorithm">Thomas algorithm</a> with a resulting computational complexity of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n^{2.5})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(n^{2.5})}</annotation>

</semantics>

</math><img src="./ace57227cac24373d5d91e9fbb5ef6ebdbaadce2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.311ex; height:3.176ex;" alt="{\displaystyle O(n^{2.5})}" loading="lazy">, <a href="Cyclic_reduction" title="Cyclic reduction">cyclic reduction</a>, <a href="Successive_overrelaxation" class="mw-redirect" title="Successive overrelaxation">successive overrelaxation</a> that has a complexity of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n^{1.5})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(n^{1.5})}</annotation>

</semantics>

</math><img src="./e5b9a912ab204a59f64111da7b880d4603426aff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.311ex; height:3.176ex;" alt="{\displaystyle O(n^{1.5})}" loading="lazy">, and <a href="Fast_Fourier_transform" title="Fast Fourier transform">Fast Fourier transforms</a> which is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n\log(n))}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(n\log(n))}</annotation>

</semantics>

</math><img src="./85994022c28e938772bd858cd8281328643e8b3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.54ex; height:2.843ex;" alt="{\displaystyle O(n\log(n))}" loading="lazy">. An optimal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(n)}</annotation>

</semantics>

</math><img src="./34109fe397fdcff370079185bfdb65826cb5565a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle O(n)}" loading="lazy"> solution can also be computed using <a href="Multigrid_methods" class="mw-redirect" title="Multigrid methods">multigrid methods</a>.<a href="#cite_note-4">[4]</a>

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

In <a href="Computational_fluid_dynamics" title="Computational fluid dynamics">computational fluid dynamics</a>, for the solution of an <a href="Incompressible_flow" title="Incompressible flow">incompressible flow</a> problem, the incompressibility condition acts as a constraint for the pressure. There is no explicit form available for pressure in this case due to a strong coupling of the velocity and pressure fields. In this condition, by taking the divergence of all terms in the momentum equation, one obtains the pressure Poisson equation.

For an incompressible flow this constraint is given by:

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial v_{x}}{\partial x}}+{\frac {\partial v_{y}}{\partial y}}+{\frac {\partial v_{z}}{\partial z}}=0}</annotation>

</semantics>

</math>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./704b7ad1ece77840fde455daa6d2e51e64282b5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.3ex; height:2.009ex;" alt="{\displaystyle v_{x}}" loading="lazy"> is the velocity in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> direction, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{y}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v_{y}}</annotation>

</semantics>

</math><img src="./922aa64dad09633a401e14be9b4389795835cd8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.177ex; height:2.343ex;" alt="{\displaystyle v_{y}}" loading="lazy"> is velocity in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{z}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v_{z}}</annotation>

</semantics>

</math><img src="./b553b9ecda9fc7f5c9da6f14a870e219080984e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle v_{z}}" loading="lazy"> is the velocity in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> direction. Taking divergence of the momentum equation and using the incompressibility constraint, the pressure Poisson equation is formed given by:

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}p=f(\nu ,V)}</annotation>

</semantics>

</math>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"> is the kinematic viscosity of the fluid and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> is the velocity vector.<a href="#cite_note-5">[5]</a>

The discrete Poisson's equation arises in the theory of <a href="Markov_chain" title="Markov chain">Markov chains</a>. It appears as the relative value function for the <a href="Dynamic_programming" title="Dynamic programming">dynamic programming</a> equation in a <a href="Markov_decision_process" title="Markov decision process">Markov decision process</a>, and as the <a href="Control_variate" class="mw-redirect" title="Control variate">control variate</a> for application in simulation <a href="Variance_reduction" title="Variance reduction">variance reduction</a>.<a href="#cite_note-MCSS-6">[6]</a><a href="#cite_note-CTCN-7">[7]</a><a href="#cite_note-AG07-8">[8]</a>

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

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</style><cite id="CITEREFHoffman2001" class="citation cs2">Hoffman, Joe (2001), "Chapter 9. Elliptic partial differential equations", Numerical Methods for Engineers and Scientists (2nd ed.), McGraw–Hill, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8247-0443-6</bdi></cite>.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> Golub, Gene H. and C. F. Van Loan, Matrix Computations, 3rd Ed., The Johns Hopkins University Press, Baltimore, 1996, pages 177–180.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> Cheny, Ward and David Kincaid, Numerical Mathematics and Computing 2nd Ed., Brooks/Cole Publishing Company, Pacific Grove, 1985, pages 443–448.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> CS267: Notes for Lectures 15 and 16, Mar 5 and 7, 1996, <a rel="nofollow" class="external free" href="https://people.eecs.berkeley.edu/~demmel/cs267/lecture24/lecture24.html">https://people.eecs.berkeley.edu/~demmel/cs267/lecture24/lecture24.html</a>

</li>

Fletcher, Clive A. J., Computational Techniques for Fluid Dynamics: Vol I, 2nd Ed., Springer-Verlag, Berlin, 1991, page 334–339.

</li>

<li id="cite_note-MCSS-6"><a href="#cite_ref-MCSS_6-0">^</a> S. P. Meyn and R.L. Tweedie, 2005. <a rel="nofollow" class="external text" href="http://probability.ca/MT/">Markov Chains and Stochastic Stability</a>.

Second edition to appear, Cambridge University Press, 2009.

</li>

<li id="cite_note-CTCN-7"><a href="#cite_ref-CTCN_7-0">^</a> S. P. Meyn, 2007. <a rel="nofollow" class="external text" href="http://www.meyn.ece.ufl.edu/archive/spm_files/CTCN/CTCN.html">Control Techniques for Complex Networks</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141216122500/http://www.meyn.ece.ufl.edu/archive/spm_files/CTCN/CTCN.html">Archived</a> December 16, 2014, at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Cambridge University Press, 2007.

</li>

<li id="cite_note-AG07-8"><a href="#cite_ref-AG07_8-0">^</a> Asmussen, Søren, Glynn, Peter W., 2007. "Stochastic Simulation: Algorithms and Analysis". Springer. Series: Stochastic Modelling and Applied Probability, Vol. 57, 2007.

</li>

</ol></div>

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

<ul><li>Hoffman, Joe D., Numerical Methods for Engineers and Scientists, 4th Ed., McGraw–Hill Inc., New York, 1992.</li>

<li><a href="Roland_Andrew_Sweet" title="Roland Andrew Sweet">Sweet, Roland A.</a>, SIAM Journal on Numerical Analysis, Vol. 11, No. 3 , June 1974, 506–520.</li>

<li><cite id="CITEREFPressTeukolskyVetterlingFlannery2007" class="citation book cs1">Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110811154417/http://apps.nrbook.com/empanel/index.html#pg=1053">"Section 20.4. Fourier and Cyclic Reduction Methods"</a>. Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-88068-8</bdi>. Archived from <a rel="nofollow" class="external text" href="http://apps.nrbook.com/empanel/index.html#pg=1053">the original</a> on August 11, 2011. Retrieved August 18, 2011.</cite></li></ul></div><div><div class="zim-footer">

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