Quadrature by fundamental solutions: kernel-independent layer potential evaluation for large collections of simple objects

TL;DR

This paper introduces a kernel-independent, efficient method for evaluating layer potentials on large collections of simple objects, avoiding complex quadrature rules and enabling fast simulations of elliptic boundary value problems.

Contribution

It presents a novel precomputation approach that transforms surface densities into effective sources, simplifying and accelerating boundary integral evaluations for multiple objects.

Findings

Efficient handling of nearly-singular integrals without specialized quadrature.

Successful application to large-scale 2D and 3D boundary value problems.

Compatibility with fast algorithms like FMM for large object collections.

Abstract

Well-conditioned boundary integral methods for the solution of elliptic boundary value problems (BVPs) are powerful tools for static and dynamic physical simulations. When there are many close-to-touching boundaries (eg, in complex fluids) or when the solution is needed in the bulk, nearly-singular integrals must be evaluated at many targets. We show that precomputing a linear map from surface density to an effective source representation renders this task highly efficient, in the common case where each object is "simple", ie, its smooth boundary needs only moderately many nodes. We present a kernel-independent method needing only an upsampled smooth surface quadrature, and one dense factorization, for each distinct shape. No (near-)singular quadrature rules are needed. The resulting effective sources are drop-in compatible with fast algorithms, with no local corrections nor bookkeeping. Our extensive numerical tests include 2D FMM-based Helmholtz and Stokes BVPs with up to 1000 objects (281000 unknowns), and a 3D Laplace BVP with 10 ellipsoids separated by $1/30$ of a diameter. We include a rigorous analysis for analytic data in 2D and 3D.