Planewave density interpolation methods for 3D Helmholtz boundary integral equations

Abstract

This paper introduces planewave density interpolation methods for the regularization of weakly singular, strongly singular, hypersingular, and nearly singular integral kernels present in 3D Helmholtz surface layer potentials and associated integral operators. Relying on Green’s third identity and pointwise interpolation of density functions in the form of planewaves, these methods allow layer potentials and integral operators to be expressed in terms of integrand functions that remain bounded or even more regular regardless of the location of the target point relative to the surface sources. Common challenging integrals that arise in both Nyström and boundary element discretization of boundary integral equations can then be numerically evaluated by standard quadrature rules irrespective of the kernel singularity. Closed-form and purely numerical planewave density interpolation procedures are presented in this paper, which are used in conjunction with Chebyshev-based Nyström and Galerkin boundary element methods. A variety of numerical examples, including problems of acoustic scattering involving multiple touching and even intersecting obstacles, demonstrate the capabilities of the proposed technique.