{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $h \u003e 0$ and let $\\Omega \\subset \\Z^2$ be a finite set of grid points.\nLet $\\partial \\Omega$ be the set of points of $\\Z^2 \\setminus \\Omega$ at distance $1$ from some point of $\\Omega$.\nLet $f \\colon \\Omega \\to \\R$ and $g \\colon \\partial \\Omega \\to \\R$ be given [[functions|0EPAT51DXQ]].\nThe *discrete Poisson equation* with Dirichlet boundary values is the problem of finding $u \\colon \\Omega \\cup \\partial \\Omega \\to \\R$ such that\n$$\\frac{u(i+1, j) + u(i-1, j) + u(i, j+1) + u(i, j-1) - 4u(i, j)}{h^2} = f(i, j) \\quad \\text{for all } (i, j) \\in \\Omega,$$\nand $u = g$ on $\\partial \\Omega$.\nThe left side is $h^{-2}$ times the [[discrete Laplace operator|ZN30Y1ZDB6]] of $u$ at $(i, j)$.\n\n## Remarks\n\nExample: let $\\Omega = \\set{(0, 0)}$, $h = 1$, $g = 0$ on the four neighbours and $f(0, 0) = -4$.\nThe equation reads $0 + 0 + 0 + 0 - 4u(0, 0) = -4$, so $u(0, 0) = 1$ is the solution.\n\nNon-example: the function with $u(0,0) = 0$ and $u = 0$ on $\\partial\\Omega$ satisfies the boundary condition but gives $0 \\ne -4$ on the left, so it is not a solution.\n\nFor a finite $\\Omega$, the problem has exactly one solution for all $f$ and $g$; this follows from the discrete maximum principle.\nOrdering the unknowns gives a linear system $A \\mathbf{u} = \\mathbf{b}$, where $A$ is sparse, symmetric and negative definite.\nOn a rectangle the matrix has a block tridiagonal form with $-4$ on the diagonal, before scaling by $h^{-2}$.\nThe same definition works in $\\Z^d$ with $2d$ neighbours and $-2d$ in place of $-4$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "6EEAEDWXZ2",
    "kind": "definition",
    "msc": [
      "39A14",
      "65N06",
      "35J05"
    ],
    "name": "Discrete Poisson equation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Discrete_Poisson_equation"
      }
    ],
    "tag": "T6DDZTB360",
    "uses": [
      "ZN30Y1ZDB6",
      "0EPAT51DXQ"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/T6DDZTB360/",
  "links": [
    {
      "field": "defines",
      "target": "6EEAEDWXZ2"
    },
    {
      "field": "uses",
      "target": "ZN30Y1ZDB6"
    },
    {
      "field": "uses",
      "target": "0EPAT51DXQ"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/T6DDZTB360.md",
  "msc": [
    "39A14",
    "65N06",
    "35J05"
  ],
  "name": "Discrete Poisson equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/T6D/T6DDZTB360.md",
  "status": null,
  "tag": "T6DDZTB360",
  "uri": "m2://T6DDZTB360",
  "url": "https://tamnd.github.io/math2/wiki/discrete-poisson-equation-definition/",
  "writing": {
    "tag": "T6DDZTB360",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}