Rapid evaluation of Newtonian potentials on planar domains

TL;DR

This paper introduces a high-order, efficient algorithm for evaluating Newtonian potentials on planar domains, leveraging boundary layer potentials and advanced polynomial interpolation for improved accuracy and speed.

Contribution

The paper presents a novel high-order algorithm combining Green's identity, Helsing-Ojala method, and polynomial interpolation for fast potential evaluation on unstructured meshes.

Findings

Achieves high accuracy with polynomial orders up to 20.

Demonstrates efficiency through numerical experiments.

Provides extensive justification for polynomial interpolation approach.

Abstract

The accurate and efficient evaluation of Newtonian potentials over general 2-D domains is important for the numerical solution of Poisson's equation and volume integral equations. In this paper, we present a simple and efficient high-order algorithm for computing the Newtonian potential over a planar domain discretized by an unstructured mesh. The algorithm is based on the use of Green's third identity for transforming the Newtonian potential into a collection of layer potentials over the boundaries of the mesh elements, which can be easily evaluated by the Helsing-Ojala method. One important component of our algorithm is the use of high-order (up to order 20) bivariate polynomial interpolation in the monomial basis, for which we provide extensive justification. The performance of our algorithm is illustrated through several numerical experiments.