A Lightweight, Geometrically Flexible Fast Algorithm for the Evaluation of Layer and Volume Potentials

TL;DR

This paper introduces a new fast, flexible algorithm for evaluating layer and volume potentials in complex geometries, compatible with various discretizations and robust against mesh irregularities.

Contribution

The authors develop a simple, efficient algorithm combining potential theory, NUFFT, and DMK that works with arbitrary triangulations and improves computational speed.

Findings

Computes solutions at a rate comparable to FFT per gridpoint.

Compatible with unstructured meshes and irregular geometries.

Insensitive to mesh flaws like gaps and degeneracies.

Abstract

Over the last two decades, several fast, robust, and high-order accurate methods have been developed for solving the Poisson equation in complicated geometry using potential theory. In this approach, rather than discretizing the partial differential equation itself, one first evaluates a volume integral to account for the source distribution within the domain, followed by solving a boundary integral equation to impose the specified boundary conditions. Here, we present a new fast algorithm which is easy to implement and compatible with virtually any discretization technique, including unstructured domain triangulations, such as those used in standard finite element or finite volume methods. Our approach combines earlier work on potential theory for the heat equation, asymptotic analysis, the nonuniform fast Fourier transform (NUFFT), and the dual-space multilevel kernel-splitting (DMK) framework. It is insensitive to flaws in the triangulation, permitting not just nonconforming elements, but arbitrary aspect ratio triangles, gaps and various other degeneracies. On a single CPU core, the scheme computes the solution at a rate comparable to that of the fast Fourier transform (FFT) in work per gridpoint.