Fully discrete spectral boundary integral methods for Helmholtz problems on smooth closed surfaces in \(\mathbb R^3\)
The paper is concerned with linear boundary integral equations of the second kind with a weakly singular kernel of a form which includes both the single and the double layer potentials for the Helmholtz equation. The boundary of the underlying 3D-domain is assumed to be diffeomorphic to the unit sphere, so that after corresponding transformations the boundary integral equation is posed on the unit sphere. A spectral Galerkin method with spherical polynomial basis functions is proposed for this equation. The occurring double integrals consist of an inner integral with weakly singular integrand, and an outer one with smooth integrand. A semidiscrete version is considered where only the outer integral is approximated by a quadrature formula, and a fully discrete version where both integrals are approximated. The main part of the paper is devoted to a stability and convergence analysis of the fully discrete method.
scientific article; zbMATH DE number 1837038
scientific article