General-purpose kernel regularization of boundary integral equations via density interpolation

TL;DR

This paper introduces a versatile high-order kernel regularization method for boundary integral equations that simplifies density interpolation without high derivatives, applicable across various PDEs and dimensions, enhancing accuracy and efficiency.

Contribution

The proposed technique offers a kernel- and dimension-independent density interpolation method that avoids explicit high-order derivatives, improving the regularization of boundary integral equations for multiple PDEs.

Findings

Demonstrates high accuracy and efficiency in 3D numerical examples.

Compatible with fast solvers for large-scale problems.

Applicable to various PDEs like Laplace, Helmholtz, and elastodynamics.

Abstract

This paper presents a general high-order kernel regularization technique applicable to all four integral operators of Calder\'on calculus associated with linear elliptic PDEs in two and three spatial dimensions. Like previous density interpolation methods, the proposed technique relies on interpolating the density function around the kernel singularity in terms of solutions of the underlying homogeneous PDE, so as to recast singular and nearly singular integrals in terms of bounded (or more regular) integrands. We present here a simple interpolation strategy which, unlike previous approaches, does not entail explicit computation of high-order derivatives of the density function along the surface. Furthermore, the proposed approach is kernel- and dimension-independent in the sense that the sought density interpolant is constructed as a linear combination of point-source fields, given by the same Green's function used in the integral equation formulation, thus making the procedure applicable, in principle, to any PDE with known Green's function. For the sake of definiteness, we focus here on Nystr\"om methods for the (scalar) Laplace and Helmholtz equations and the (vector) elastostatic and time-harmonic elastodynamic equations. The method's accuracy, flexibility, efficiency, and compatibility with fast solvers are demonstrated by means of a variety of large-scale three-dimensional numerical examples.