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

This paper presents a general high-order kernel regularization technique applicable to all four integral operators of Calderon 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. (C) 2021 Elsevier B.V. All rights reserved.

Automated summary

This paper introduces a general high-order kernel regularization technique that can be applied to linear elliptic PDEs in two and three spatial dimensions. By interpolating the density function and solutions of the underlying homogeneous PDE, singular and nearly singular integrals are converted into bounded integrands without explicit computation of high-order derivatives. The proposed approach is kernel- and dimension-independent, showing accuracy, flexibility, efficiency, and compatibility with fast solvers through large-scale three-dimensional numerical examples.