Discrete Poisson equation

In mathematics, the discrete Poisson equation is the finite difference analog of the Poisson equation. In it, the discrete Laplace operator takes the place of the Laplace operator. The discrete Poisson equation is frequently used in numerical analysis as a stand-in for the continuous Poisson equation, although it is also studied in its own right as a topic in discrete mathematics. Using the finite difference numerical method to discretizethe 2-dimensional Poisson equation (assuming a uniform spatial discretization, Δ x = Δ y {displaystyle Delta x=Delta y} ) on an m × n grid gives the following formula: where 2 ≤ i ≤ m − 1 {displaystyle 2leq ileq m-1} and 2 ≤ j ≤ n − 1 {displaystyle 2leq jleq n-1} . The preferred arrangement of the solution vector is to use natural ordering which, prior to removing boundary elements, would look like: This will result in an mn × mn linear system: