High-order kernel regularization of singular and hypersingular Helmholtz boundary integral operators

TL;DR

This paper develops a high-order kernel regularization method for all Helmholtz boundary integral operators in 3D, enabling accurate, simple-to-implement solutions for scattering problems with explicit convergence rates.

Contribution

It introduces the first high-order regularization for the hypersingular Helmholtz operator in 3D, combining error analysis with practical implementation advantages.

Findings

Achieves explicit convergence rates depending on regularization order and quadrature degree.

Simplifies implementation by reducing to smooth integral evaluations without specialized quadrature.

Demonstrates effectiveness through numerical examples on scattering problems.

Abstract

This paper extends and analyzes the high-order kernel regularization framework of Beale & Tlupova (arXiv:2510.13639) to all four on-surface boundary integral operators of the Helmholtz Calderon calculus in three dimensions: the single-layer, double-layer, adjoint double-layer, and hypersingular operators. To the best of our knowledge, this work provides the first high-order kernel regularization of the hypersingular operator for both the Helmholtz and Laplace equations in three dimensions. The regularization replaces each singular kernel with a smooth modification constructed from error functions together with a polynomial correction whose coefficients are determined through moment conditions. Alongside the derivation of the regularizing functions, the paper provides a unified error analysis of the combined regularization and quadrature discretization procedure. By coupling the regularization parameter to the mesh size, the two error contributions can be balanced, leading to explicit overall convergence rates that depend jointly on the order of the regularization and the degree of exactness of the surface quadrature rule. A key practical feature of the method is its implementation simplicity. Once the regularizing functions are determined, the numerical task reduces entirely to the evaluation of smooth surface integrals using standard quadrature, without the need for element-local solves, singularity-specific precomputations, or specialized quadrature rules. Although the modified kernel is generally incompatible with kernel-specific fast methods, this limitation is addressed through H-matrix acceleration, applicable in a black-box manner. Numerical examples -- including verification of the predicted convergence rates and solution of sound-soft and sound-hard scattering problems by smooth obstacles -- demonstrate the accuracy and practicality of the proposed methodology.