Convergence analysis of the finite difference ADI scheme for the heat equation on a convex set

Abstract

It is well known that for the heat equation on a rectangle, the finite difference alternating direction implicit (ADI) method converges with order two. For the first time in the literature, we bound errors of the finite difference ADI method for the heat equation on a convex set for which it is possible to construct a partition consistent with the boundary. Numerical results indicate that the ADI method may also work for some nonconvex sets for which it is possible to construct a partition consistent with the boundary.