A symmetric Galerkin BEM variational framework for multi-domain interface problems

Abstract

In this paper we discuss an energy-based variational framework for the solution of interior problems in multiply-connected domains comprising multiple piecewise homogeneous subdomains, using exclusively boundary integral equations. The primary goal is to provide a unified variational setting that lends itself naturally to symmetric Galerkin boundary element formulations in terms of Dirichlet-type variables only. The approach hinges on the explicit imposition of the normal derivative of the classical integral representation of the interior solution on each subdomain via Lagrange multipliers in the augmented Lagrangian of the system. We use Maue-type identities to resolve the hypersingular kernels, leading to a scheme that requires only standard single- and double-layer evaluations. In addition, the usual difficulty with multi-valued normals at subdomain corners is treated here within the same variational framework, by incorporating into the variational formulation the constraint equation between the limiting normal derivatives at either side of the corner. The resulting scheme remains fully symmetric. The numerical implementation avoids the explicit presence of Neumann-type unknowns on the boundaries, through condensation at the subdomain level. In all integral evaluations, three- or four-point Gauss quadrature rules are sufficient for accurate results. We describe the theory and present illustrative examples for thermal and acoustic problems governed by Laplace and Helmholtz equations, respectively. This technique, however, can be applied without essential modification to more general problems.