Report Number: CS-TR-70-159
Institution: Stanford University, Department of Computer Science
Title: The use of direct methods for the solution of the discrete Poisson equation on non-rectangular regions
Author: George, John Alan
Date: June 1970
Abstract: Some direct and iterative schemes are presented for solving a standard finite-difference scheme for Poisson's equation on a two-dimensional bounded region R with Dirichlet conditions specified on the boundary $\delta$R. These procedures make use of special-purpose direct methods for solving rectangular Poisson problems. The region is imbedded in a rectangle and a uniform mesh is superimposed on it. The usual five-point Poisson difference operator is applied over the whole rectangle, yielding a block-tridiagonal system of equations. The original problem, however, determines only the elements of the right-hand side which correspond to grid points lying within $\delta$R; the remaining elements can be treated as parameters. The iterative algorithms construct a sequence of right-hand sides in such a way that the corresponding sequence of solutions on the rectangle converges to the solution of the imbedded problem.