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.

http://i.stanford.edu/pub/cstr/reports/cs/tr/70/159/CS-TR-70-159.pdf