A fast method for evaluating Volume potentials in the Galerkin boundary element method

TL;DR

This paper introduces three algorithms that efficiently evaluate volume potentials in boundary element methods for elliptic PDEs, achieving nearly linear complexity and customizable accuracy, demonstrated on 3D Poisson problems.

Contribution

The paper presents a novel application of a modified fast multipole method to boundary-concentrated volume meshes, reducing computational complexity to match boundary potential calculations.

Findings

Algorithms achieve nearly O(h^{-2}) complexity.

Potential errors can be controlled to converge at any rate O(h^p).

Demonstrated effectiveness on 3D Poisson equation potentials.

Abstract

Three algorithm are proposed to evaluate volume potentials that arise in boundary element methods for elliptic PDEs. The approach is to apply a modified fast multipole method for a boundary concentrated volume mesh. If $h$ is the meshwidth of the boundary, then the volume is discretized using nearly $O(h^{-2})$ degrees of freedom, and the algorithm computes potentials in nearly $O(h^{-2})$ complexity. Here nearly means that logarithmic terms of $h$ may appear. Thus the complexity of volume potentials calculations is of the same asymptotic order as boundary potentials. For sources and potentials with sufficient regularity the parameters of the algorithm can be designed such that the error of the approximated potential converges at any specified rate $O(h^p)$. The accuracy and effectiveness of the proposed algorithms are demonstrated for potentials of the Poisson equation in three dimensions.