Fredrik Fryklund: A Fast Algorithm for the Evaluation of Layer and Volume Potentials

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, I will 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. The approach combines earlier work on potential theory for the heat equation, asymptotic analysis, and the nonuniform fast Fourier transform (NUFFT). It is insensitive to flaws in the triangulation, permitting not just nonconforming elements, but arbitrary aspect ratio triangles, gaps and various others degeneracies.