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Recursive reduction quadrature for the evaluation of Laplace layer potentials in three dimensions

Shidong Jiang, Hai Zhu

Abstract

A high-order quadrature scheme is constructed for the evaluation of Laplace single and double layer potentials and their normal derivatives on smooth surfaces in three dimensions. The construction begins with a harmonic approximation of the density on each patch, which allows for a natural harmonic polynomial extension in a volumetric neighborhood of the patch in the ambient space. Then by the general Stokes theorem, singular and nearly singular surface integrals are reduced to line integrals preserving the singularity of the kernel, instead of the standard origin-centered 1-forms that require expensive adaptive integration. These singularity-preserving line integrals can be semi-analytically evaluated using singularity-swap quadrature. In other words, the evaluation of singular and nearly singular surface integrals is reduced to function evaluations at the vertices on the boundary of each patch. The recursive reduction quadrature largely removes adaptive integration that is needed in most existing high-order quadratures for singular and nearly singular surface integrals, resulting in exceptional performance. The scheme achieves twelve-digit accuracy uniformly for close evaluations and offers a speedup of five times or more in constructing the sparse quadrature-correction matrix compared to previous state-of-the-art quadrature schemes.

Recursive reduction quadrature for the evaluation of Laplace layer potentials in three dimensions

Abstract

A high-order quadrature scheme is constructed for the evaluation of Laplace single and double layer potentials and their normal derivatives on smooth surfaces in three dimensions. The construction begins with a harmonic approximation of the density on each patch, which allows for a natural harmonic polynomial extension in a volumetric neighborhood of the patch in the ambient space. Then by the general Stokes theorem, singular and nearly singular surface integrals are reduced to line integrals preserving the singularity of the kernel, instead of the standard origin-centered 1-forms that require expensive adaptive integration. These singularity-preserving line integrals can be semi-analytically evaluated using singularity-swap quadrature. In other words, the evaluation of singular and nearly singular surface integrals is reduced to function evaluations at the vertices on the boundary of each patch. The recursive reduction quadrature largely removes adaptive integration that is needed in most existing high-order quadratures for singular and nearly singular surface integrals, resulting in exceptional performance. The scheme achieves twelve-digit accuracy uniformly for close evaluations and offers a speedup of five times or more in constructing the sparse quadrature-correction matrix compared to previous state-of-the-art quadrature schemes.

Paper Structure

This paper contains 26 sections, 14 theorems, 136 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Mathematical apparatus
  3. Quaternion algebra
  4. Differential forms and Stokes theorem
  5. Harmonic polynomials
  6. Singularity-swap quadrature
  7. Summary of main analytic results in zhu2021thesiszhu2022sisc
  8. Main results
  9. Summary of improvements made over the work in zhu2021thesiszhu2022sisc
  10. Choice of basis functions for the $p$th order harmonic approximation
  11. New treatment of the Laplace single layer potential
  12. Singularity preserving 1-forms
  13. DLP 1-forms for constant density
  14. 1-to-0 form conversion
  15. Evaluation of the normal derivative of the single layer potential
  16. ...and 11 more sections

Key Result

Lemma 2.1

\newlabellemma10 Suppose that $P$ is a contractible surface element in $\mathbb{R}^3$ and that $f$ is a real-valued function defined in a neighborhood of $P$ in $\mathbb{R}^3$. Then the differential 2-form $\alpha={\boldsymbol f}({\boldsymbol r})\cdot \boldsymbol{\nu}({\boldsymbol r}) da({\boldsym satisfies the equation and thus In other words, the surface integral $\int_P \alpha$ is reduced to

Figures (8)

  • Figure 1: Left: triangulated warped torus boundary. Right: relative $l_\infty$ errors as functions of $N_{\rm patches}$ for various orders $p$.
  • Figure 2: Slice plots of $\log_{10}$ of the pointwise relative errors for the warped torus. Left: $p=8,\ n_{\theta} = 36,\ n_{\phi} = 72$. Right: $p=14,\ n_{\theta} = 48,\ n_{\phi} = 96$.
  • Figure 3: Left: triangulated cushion-shaped boundary. Right: relative $l_\infty$ errors as functions of $N_{\rm patches}$ for various orders $p$.
  • Figure 4: Slice plots of $\log_{10}$ of the pointwise relative errors for the cushion-shape boundary. Left: $p=6, n_{\theta} = 17, n_{\phi} = 17$. Right: $p=12, n_{\theta} = 33, n_{\phi} = 33$.
  • Figure 5: Triangulated stellarator-shaped geometry.
  • ...and 3 more figures

Theorems & Definitions (36)

  • Lemma 2.1
  • Proof 1
  • Remark 2.2
  • Lemma 2.3
  • Theorem 3.1
  • Lemma 3.2
  • Proof 2
  • Lemma 3.3
  • Proof 3
  • Lemma 3.4
  • ...and 26 more