Ubiquitous evaluation of layer potentials using Quadrature by Kernel-Independent Expansion

TL;DR

This paper presents QBKIX, a kernel-independent quadrature scheme for high-order accurate evaluation of layer potentials near and on domain boundaries, applicable to various elliptic PDEs without requiring analytic expansions.

Contribution

The paper introduces QBKIX, a novel kernel-independent quadrature method that simplifies implementation and extends to multiple kernels and dimensions, improving upon existing techniques.

Findings

Accurately evaluates layer potentials for multiple PDE kernels.

Compatible with fast algorithms like kernel-independent FMM.

Effective on smooth and corner domains in 2D.

Abstract

We introduce a quadrature scheme--QBKIX--for the high-order accurate evaluation of layer potentials associated with general elliptic PDEs near to and on the domain boundary. Relying solely on point evaluations of the underlying kernel, our scheme is essentially PDE-independent; in particular, no analytic expansion nor addition theorem is required. Moreover, it applies to boundary integrals with singular, weakly singular, and hypersingular kernels. Our work builds upon Quadrature by Expansion (QBX), which approximates the potential by an analytic expansion in the neighborhood of each expansion center. In contrast, we use a sum of fundamental solutions lying on a ring enclosing the neighborhood, and solve a small dense linear system for their coefficients to match the potential on a smaller concentric ring. We test the new method with Laplace, Helmholtz, Yukawa, Stokes, and Navier (elastostatic) kernels in two dimensions (2D) using adaptive, panel-based boundary quadratures on smooth and corner domains. Advantages of the algorithm include its relative simplicity of implementation, immediate extension to new kernels, dimension-independence (allowing simple generalization to 3D), and compatibility with fast algorithms such as the kernel-independent FMM.