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A Simple Method for Computing Singular or Nearly Singular Integrals on Closed Surfaces

Part of: Elliptic equations and systems Integral equations, integral transforms Higher-dimensional theory Numerical approximation and computational geometry

Published online by Cambridge University Press:  31 August 2016

J. Thomas Beale ,
Wenjun Ying  and
Jason R. Wilson
J. Thomas Beale*
Affiliation:
Department of Mathematics, Duke University, Durham, NC 27708-0320, USA
Wenjun Ying*
Affiliation:
Department of Mathematics, MOE-LSC and Institute of Natural Sciences, Shanghai Jiao Tong University, Minhang, Shanghai 200240, P.R. China
Jason R. Wilson*
Affiliation:
Mathematics Department, Virginia Tech, Blacksburg, VA 24061-0123, USA
*
*Corresponding author. Email addresses:beale@math.duke.edu (J. T. Beale), wying@sjtu.edu.cn (W. Ying), jasonwil@math.vt.edu (J. R. Wilson)
*Corresponding author. Email addresses:beale@math.duke.edu (J. T. Beale), wying@sjtu.edu.cn (W. Ying), jasonwil@math.vt.edu (J. R. Wilson)
*Corresponding author. Email addresses:beale@math.duke.edu (J. T. Beale), wying@sjtu.edu.cn (W. Ying), jasonwil@math.vt.edu (J. R. Wilson)
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Abstract

We present a simple, accurate method for computing singular or nearly singular integrals on a smooth, closed surface, such as layer potentials for harmonic functions evaluated at points on or near the surface. The integral is computed with a regularized kernel and corrections are added for regularization and discretization, which are found from analysis near the singular point. The surface integrals are computed from a new quadrature rule using surface points which project onto grid points in coordinate planes. The method does not require coordinate charts on the surface or special treatment of the singularity other than the corrections. The accuracy is about O(h3), where h is the spacing in the background grid, uniformly with respect to the point of evaluation, on or near the surface. Improved accuracy is obtained for points on the surface. The treecode of Duan and Krasny for Ewald summation is used to perform sums. Numerical examples are presented with a variety of surfaces.

Keywords

Boundary integral methodlayer potentialnearly singular integralsLaplace equationsurface integralimplicit surface

MSC classification

Secondary: 65R20: Integral equations 65D30: Numerical integration 31B10: Integral representations, integral operators, integral equations methods 35J08: Green's functions
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 3 , September 2016 , pp. 733 - 753
Copyright
Copyright © Global-Science Press 2016 

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