Solution of poisson equations on a nonuniform grid

Abstract The solution of a Poisson equation expressed in finite-difference form on a nonuniform multidimensional mesh of gridpoints in an orthogonal coordinate system is discussed. The discussion is specifically directed toward problems which require the solution of many such equations with the gridpoint distribution and boundary conditions fixed, which implies that in comparing different techniques the task of generating required constants may be neglected. Extension of the matrix decomposition technique to cover the case of smoothly varying grid-intervals is considered when zero-gradient, zero-value, or (under certain conditions) periodic boundary conditions are incorporated. This method is compared in particular with the widely used Alternating Direction Implicit procedure, and it is concluded that for sufficiently small meshes, not exceeding about 40 points in any direction, it is both faster and more precise than ADI.