A boundary element method for the Helmholtz equation using finite part integration

Abstract

The classical boundary integral equation formulation of the Helmholtz equation in the exterior domain, derived from Green's second theorem, does not have a unique solution for certain real values of the wavenumber. The use of the Burton and Miller formulation to ensure a unique solution introduces certain computational difficulties in the form of finite part integrals. In this past these difficulties have been overcome by the use of a simple piecewise constant approximation to the solution. In this paper special quadrature rules are developed for evaluating the finite part integrals, demonstrating that it is possible to employ a higher-order approximation, such as piecewise quadratic, to the solution. The numerical results show that it is possible to obtain an accurate solution to a wide range of problems using these techniques.