Numerical Implementation of the Schwarz Alternating Procedure for Elliptic Partial Differential Equations

Abstract

The Schwarz alternating procedure provides a method for the numerical solution of certain boundary value problems. The region must consist of the overlapping union of two or more simple regions, and solutions to the boundary value problem must be particularly easy to compute for the simple regions. For example, Poisson’s integral provides an explicit formula for the solution to Laplace’s equation on a disk. Using this integral, Schwarz’s method permits a solution of Laplace’s equation on the union of two disks as a convergent alternating sequence of solutions on two disjoint disks. This paper discusses the numerical implementation for this model problem, paying particular attention to the treatment of singularities in the Poisson kernel and at the corners of the region. After demonstrating the speed and accuracy of the method for this model problem, the techniques are used to compute the capacity of a lens to six significant digits. This is a classical problem of long-standing interest, for which an exact answer is not known. The best previously reported value, computed by another technique, has three significant digits.