Abstract
In this paper we review, analyse and discuss several boundary integral formulations for a stable solution of exterior boundary value and transmission problems for the Helmholtz equation. Based on the characterisation of spurious modes, which correspond to eigensolutions of the interior Laplace equation, direct, indirect, combined and regularised boundary integral equations are considered for the solution of the exterior Dirichlet boundary value problem. In addition to established approaches, such as Burton/Miller, Brakhage/Werner, or the CHIEF method, we also include a discussion of more recent results which can be applied also in the more general case of Lipschitz domains. For a stable boundary integral formulation for the solution of transmission problems, we rely on the use of both boundary integral equations of the direct approach, and on suitable linear combinations. Here we restrict ourselves to single trace formulations, including the rather standard Steklov–Poincare operator formulation. This contribution reviews the mathematical analysis of stable boundary integral formulations for the solution of Helmholtz boundary value and transmission problems, and it will provide a foundation for the error and stability analysis of related Galerkin boundary element methods.
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