WS,Leslie Greengard: Lightweight, geometrically flexible algorithms for the evaluation of layer and volume potentials

Speaker
Leslie Greengard,New York University

Abstract
Over the last two decades, a variety of fast, robust, and high-order accurate methods have been developed for solving elliptic and parabolic PDEs 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.

We present a new set of algorithms 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 FFT in work per gridpoint.

This is joint work with Fredrik Fryklund, Shidong Jiang and Jun Wang.