On the numerical solution of Helmholtz’s equation by the capacitance matrix method

In recent years the usefulness of fast Laplace solvers has been extended to problems on arbitrary regions in the plane by the development of capacitance matrix methods. The solution of the Dirichlet and Neumann problems for Helmholtz’s equation is considered. It is shown, that by an appropriate choice of the fast solver, the capacitance matrix can be generated quite inexpensively. An analogy between capacitance matrix methods and classical potential theory for the solution of Laplace’s equation is explored. This analogy suggests a modification of the method in the Dirichlet case. This new formulation leads to well-conditioned capacitance matrix equations which can be solved quite efficiently by the conjugate gradient method. A highly accurate solution can, therefore, be obtained at an expense which grows no faster than that for a fast Laplace solver on a rectangle when the mesh size is decreased.