Laplace and Helmholtz Problems

Consider the Poisson equation −∆u = f in a planar exterior domain of a bounded domain O. Assume that f = 0 in the outside of a disc with sufficiently large diameter. The solution u is assumed to be bounded at infinity. Discretizing the problem, we employ the finite element method (FEM, in short) inside the disc, and the charge simulation method (CSM, in short) outside the disc. A result of mathematical analysis for this FEM-CSM combined method is reported in this paper. CSM is a typical example of the fundamental solution method (FSM, in short), through which the solution of homogeneous partial differential equation is approximated as a linear combination of fundamental solutions of differential operator. Hence the combined method for 2D exterior Laplace problem is extendable to the planar exterior reduced wave equations. Our discretization procedure for the reduced wave equation is also described in the paper.