A new study of the Burton and Miller method for the solution of a 3D Helmholtz problem

Abstract

The exterior Helmholtz problem can be efficiently solved by reformulating the differential equation as an integral equation over the surface of the radiating and/or scattering object. One popular approach for overcoming either non-unique or non-existent problems which occur at certain values of the wave number is the so-called Burton and Miller method which modifies the usual integral equation into one which can be shown to have a unique solution for all real and positive wave numbers. This formulation contains an integral operator with a hypersingular kernel function and for many years, a commonly used method for overcoming this hypersingularity problem has been the collocation method with piecewise-constant polynomials. Viable high-order methods only exist for the more expensive Galerkin method. This paper proposes a new reformulation of the Burton–Miller approach and enables the more practical collocation method to be applied with any high-order piecewise polynomials. This work is expected to lead to much progress in subsequent development of fast solvers. Numerical experiments on 3D domains are included to support the proposed high-order collocation method.