Fast, high-order numerical evaluation of volume potentials via polynomial density interpolation

TL;DR

This paper introduces a high-order numerical method for efficiently evaluating singular volume integrals related to Poisson and Helmholtz equations in 2D, reducing computational complexity while maintaining accuracy.

Contribution

It develops a novel polynomial density interpolation approach that simplifies singular integral evaluation by separating regularized volume integrals from layer potentials.

Findings

Achieves high-order accuracy in volume potential evaluation.

Reduces the need for specialized quadrature for singularities.

Compatible with fast algorithms for efficient computation.

Abstract

This article presents a high-order accurate numerical method for the evaluation of singular volume integral operators, with attention focused on operators associated with the Poisson and Helmholtz equations in two dimensions. Following the ideas of the density interpolation method for boundary integral operators, the proposed methodology leverages Green's third identity and a local polynomial interpolant of the density function to recast the volume potential as a sum of single- and double-layer potentials and a volume integral with a regularized (bounded or smoother) integrand. The layer potentials can be accurately and efficiently evaluated everywhere in the plane by means of existing methods (e.g. the density interpolation method), while the regularized volume integral can be accurately evaluated by applying elementary quadrature rules. Compared to straightforwardly computing corrections for every singular and nearly-singular volume target, the method significantly reduces the amount of required specialized quadrature by pushing all singular and near-singular corrections to near-singular layer-potential evaluations at target points in a small neighborhood of the domain boundary. Error estimates for the regularization and quadrature approximations are provided. The method is compatible with well-established fast algorithms, being both efficient not only in the online phase but also to set-up. Numerical examples demonstrate the high-order accuracy and efficiency of the proposed methodology; applications to inhomogeneous scattering are presented.