# Planewave density interpolation methods for 3D Helmholtz boundary integral equations

## TL;DR

This paper presents planewave density interpolation methods that regularize singular and nearly singular integral kernels in 3D Helmholtz boundary integral equations, enabling accurate numerical evaluation regardless of target position.

## Contribution

The paper introduces novel planewave density interpolation techniques for regularizing integral kernels in 3D Helmholtz equations, compatible with Nyström and boundary element methods.

## Key findings

- Methods effectively handle singular and nearly singular integrals.
- Numerical examples demonstrate accuracy in acoustic scattering problems.
- Compatible with standard quadrature rules for improved computation.

## 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 smooth (at least bounded) regardless the location of the target point relative to the surface sources. Common challenging integrals that arise in both Nystr\"om and boundary element discretization of boundary integral equation, can then be numerically evaluated by standard quadrature rules that are 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 Nystr\"om 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.