Discrete Poisson equation

Summary

The discrete Poisson equation $\Delta u = f$ asks for a function on a grid or graph whose discrete Laplacian is a given function $f$. It is the finite difference version of Poisson's equation $\Delta u = f$, and with $f = 0$ it becomes the discrete form of Laplace's equation. With boundary values prescribed, it becomes a large sparse linear system. Solving this system fast is a central task of numerical analysis, done with iterative methods, multigrid or the fast Fourier transform.