Boundary integral equations for mixed Dirichlet, Neumann and transmission problems

Abstract

This paper presents a theoretical discussion as well as novel solution algorithms for problems of scattering on smooth two-dimensional domains under Zaremba boundary conditions-for which Dirichlet and Neumann conditions are specified on various portions of the domain boundary. The theoretical basis of the proposed numerical methods, which is provided for the first time in the present contribution, concerns detailed information about the singularity structure of solutions of the Helmholtz operator under boundary conditions of Zaremba type. The new numerical method is based on use of Green functions and integral equations, and it relies on the Fourier Continuation method for regularization of all smooth-domain Zaremba singularities as well as newly derived quadrature rules which give rise to high-order convergence even around Zaremba singular points. As demonstrated in this paper, the resulting algorithms enjoy high-order convergence, and they can be used to efficiently solve challenging Helmholtz boundary value problems and Laplace eigenvalue problems with high-order accuracy.

Galerkin Boundary Element Methods for Electromagnetic Scattering

“…Taking our cue from the approach to acoustic scattering in [61], we introduce scaled boundary integral operators…”

Direct boundary integral equation method for electromagnetic scattering by partly coated dielectric objects

“…The next result promotes Calderon projectors to a pivotal role in the derivation of boundary integral equations, c.f. [35,Theorem 3.7]. …”

“…This insight made it possible to come up with new formulations for coupled acoustic and electromagnetic scattering problems [9,23,35]. First, we use the bottom equations in (25) and (26), and get…”

“…The proof of existence of solutions for (35) hinges on a generalized Gårding inequality satisfied by the bilinear form underlying (35) and employs the Fredholm alternative. The technique is elaborated in [8,Sect.…”

“…We aim to use a conforming Galerkin boundary element discretization of (35). To that end, will be approximated by a triangulation h composed of flat triangles.…”