Quadrature by Expansion: A New Method for the Evaluation of Layer Potentials

TL;DR

This paper introduces Quadrature by Expansion (QBX), a high-order, systematic method for accurately evaluating boundary integrals with singular kernels, compatible with fast algorithms and applicable to various PDE boundary problems.

Contribution

QBX provides a unified, high-order approach for singular integral evaluation that is easy to implement and works with complex geometries and fast algorithms.

Findings

QBX achieves high accuracy for singular and hypersingular kernels.

The method is compatible with fast hierarchical algorithms like the fast multipole method.

Numerical tests confirm QBX's effectiveness on smooth and corner domains.

Abstract

Integral equation methods for the solution of partial differential equations, when coupled with suitable fast algorithms, yield geometrically flexible, asymptotically optimal and well-conditioned schemes in either interior or exterior domains. The practical application of these methods, however, requires the accurate evaluation of boundary integrals with singular, weakly singular or nearly singular kernels. Historically, these issues have been handled either by low-order product integration rules (computed semi-analytically), by singularity subtraction/cancellation, by kernel regularization and asymptotic analysis, or by the construction of special purpose "generalized Gaussian quadrature" rules. In this paper, we present a systematic, high-order approach that works for any singularity (including hypersingular kernels), based only on the assumption that the field induced by the integral operator is locally smooth when restricted to either the interior or the exterior. Discontinuities in the field across the boundary are permitted. The scheme, denoted QBX (quadrature by expansion), is easy to implement and compatible with fast hierarchical algorithms such as the fast multipole method. We include accuracy tests for a variety of integral operators in two dimensions on smooth and corner domains.