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Quadrature for Singular Integrals over convex Polytopes

Johannes Tausch

TL;DR

The paper tackles the problem of efficiently evaluating weakly singular integrals over general convex polytopes, central to Galerkin boundary element and nonlocal PDE discretizations. It introduces a general pyramidal decomposition that splits a polytope into convex hulls (conv$(A,B)$) of singular and regular faces, transforming the singularity into a single parameter $\lambda$ that is accurately integrated with Gauss-Jacobi quadrature. The framework applies to Cartesian products of polytopes, with explicit decompositions provided for the Cartesian products of two simplices and two cubes, along with comparative analyses against prior methods. Numerical experiments on simplex products demonstrate exponential convergence for analytic kernels and illustrate the practical efficiency of generalized Gauss rules versus tensor-product quadrature. This approach generalizes singular quadrature to arbitrary polytopes and holds significant potential for improving accuracy and efficiency in boundary integral methods and related applications.

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.

Quadrature for Singular Integrals over convex Polytopes

TL;DR

The paper tackles the problem of efficiently evaluating weakly singular integrals over general convex polytopes, central to Galerkin boundary element and nonlocal PDE discretizations. It introduces a general pyramidal decomposition that splits a polytope into convex hulls (conv) of singular and regular faces, transforming the singularity into a single parameter that is accurately integrated with Gauss-Jacobi quadrature. The framework applies to Cartesian products of polytopes, with explicit decompositions provided for the Cartesian products of two simplices and two cubes, along with comparative analyses against prior methods. Numerical experiments on simplex products demonstrate exponential convergence for analytic kernels and illustrate the practical efficiency of generalized Gauss rules versus tensor-product quadrature. This approach generalizes singular quadrature to arbitrary polytopes and holds significant potential for improving accuracy and efficiency in boundary integral methods and related applications.

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.

Paper Structure

This paper contains 18 sections, 8 theorems, 82 equations, 6 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Convex Polytopes
  3. Basic Concepts
  4. Iterated Pyramids
  5. Convex Hull of two Polytopes
  6. Distance Formula
  7. Integrals over Faces and Convex Hulls
  8. Pyramidal Decomposition Algorithm
  9. Decomposition Algorithm
  10. An Example
  11. Triangulations
  12. Singular Kernel Integrals
  13. Examples
  14. Cartesian Product of Two Simplices
  15. Cartesian Product of Two Cubes
  16. ...and 3 more sections

Key Result

Lemma 1

If Assumption asu:convAB holds, then $\mathbf{x}_A \in A$ and $\mathbf{x}_B \in B$ and $\lambda \in [0,1]$ are unique.

Figures (6)

  • Figure 1: Double pyramid with singular vertices $\mathbf{v}_0,\mathbf{v}_1,\mathbf{v}_2,\mathbf{v}_3$ and regular vertices $\mathbf{v}_4$ and $\mathbf{v}_5$ (left) and the resulting pyramidal lattice. The first index in a face denotes the apex for the next pyramid.
  • Figure 1: Pyramidal decomposition of a triangle and a rectangle with a common edge. The apex of the next base is always listed first.
  • Figure 1: Vertices and faces of $S$ and $T$.
  • Figure 1: Quadrature errors vs. the order (left) and the number of function evaluations. Self case of the three and five dimensional simplex.
  • Figure 2: Pyramidal lattice for the cube.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Lemma 1
  • Proof 1
  • Lemma 2
  • Proof 2
  • Lemma 1
  • Proof 3
  • Lemma 1
  • Proof 4
  • Lemma 2
  • Proof 5
  • ...and 3 more