Fractional Laplacian - Quadrature rules for singular double integrals in 3D

TL;DR

This paper develops specialized quadrature rules using Duffy transformation for efficiently computing the fractional Laplacian's stiffness matrix in 3D, ensuring accuracy comparable to finite element errors.

Contribution

It introduces adapted quadrature rules for 3D fractional Laplacian integrals, providing error bounds and demonstrating adaptability to other singular double integrals.

Findings

Quadrature rules based on Duffy transformation for 3D fractional Laplacian.

Error bounds matching finite element error magnitude.

Method adaptable to other 3D singular integrals.

Abstract

In this article, quadrature rules for the efficient computation of the stiffness matrix for the fractional Laplacian in three dimensions are presented. These rules are based on the Duffy transformation, which is a common tool for singularity removal. Here, this transformation is adapted to the needs of the fractional Laplacian in three dimensions. The integrals resulting from this Duffy transformation are regular integrals over less-dimensional domains. We present bounds on the number of Gauss points to guarantee error estimates which are of the same order of magnitude as the finite element error. The methods presented in this article can easily be adapted to other singular double integrals in three dimensions with algebraic singularities.