A tensor product generalized ADI method for elliptic problems on cylindrical domains with holes

摘要

We consider solving second order linear elliptic partial differential equations together with Dirichlet boundary conditions in three dimensions on cylindrical domains (nonrectangular in x and y) with holes.We approximate the partial differential operators by standard partial difference operators. If the partial differential operator separates into two terms, one depending on x and y, and one depending on z, then the discrete elliptic problem may be written in tensor product form as (Tz⊗I + I⊗Axy) U=F. We consider a specific implementation which uses a Method of Planes approach with unequally spaced finite differences in the xy direction and symmetric finite difference in the z direction. We establish the convergence of the Tensor Product Generalized Alternating Direction Implicit iterative method applied to such discrete problems. We show that this method gives a fast and memory efficient scheme for solving a large class of elliptic problems.