Indirect methods with Brakhage-Werner potentials for Helmholtz transmission problems.

*(English)* Zbl 1119.65414
Bermúdez de Castro, Alfredo (ed.) et al., Numerical mathematics and advanced applications. Proceedings of ENUMATH 2005, the 6th European conference on numerical mathematics and advanced applications, Santiago de Compostela, Spain, July 18–22, 2005. Berlin: Springer (ISBN 3-540-34287-7/hbk). 1146-1154 (2006).

Summary: We propose and analyse numerical methods for Helmholtz transmission problems in two and three dimensions. The methods we analyse use combined single and double layer potentials to represent the interior and exterior solution of the transmission problem. The corresponding boundary integral system includes weakly singular and hypersingular boundary integral operators on the interface. Its invertibility is equivalent to the unique solvability of the transmission problem, since the use of the above mentioned potentials does not introduce spurious eigenmodes in the formulation. We give necessary and sufficient conditions for the convergence of general Petrov-Galerkin schemes for solving the resulting system, providing some concrete methods for the two-dimensional case. Some numerical experiments are shown.

##### MSC:

65N38 | Boundary element methods (BVP of PDE) |

35J05 | Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation |

65N12 | Stability and convergence of numerical methods (BVP of PDE) |