Quadrature for Singular Integrals over convex Polytopes

TL;DR

This paper introduces an efficient numerical algorithm for approximating weakly singular integrals over convex polytopes, leveraging decomposition and Gauss-Jacobi quadrature to handle singularities effectively.

Contribution

It presents a novel decomposition-based quadrature method for singular integrals over convex polytopes, applicable to general finite polytopes and demonstrated with numerical examples.

Findings

Effective convergence of the quadrature scheme demonstrated

Applicable to general convex polytopes including simplices and cubes

Numerical results confirm the method's efficiency and accuracy

Abstract

A new algorithm for the efficient numerical approximation of weakly singular integrals over convex polytopes is introduced. Such integrals appear in the Galerkin discretizations of integral equations and nonlocal partial differential equations. The polytope is decomposed into a number of convex hulls of a singular and regular face. This expresses the singularity in a single variable which is effectively handled by Gauss-Jacobi quadrature. The decomposition algorithm is applicable to general finite polytopes. The Cartesian product of two simplices and two cubes will be discussed as special cases and numerical examples will be presented to illustrate the convergence of the resulting quadrature scheme.