Volume 20, Issue 3
A Simple Method for Computing Singular or Nearly Singular Integrals on Closed Surfaces

J. Thomas Beale, Wenjun Ying & Jason R. Wilson

Commun. Comput. Phys., 20 (2016), pp. 733-753.

Published online: 2018-04

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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($h^3$), 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.

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@Article{CiCP-20-733, author = {}, title = {A Simple Method for Computing Singular or Nearly Singular Integrals on Closed Surfaces}, journal = {Communications in Computational Physics}, year = {2018}, volume = {20}, number = {3}, pages = {733--753}, 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($h^3$), 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.030815.240216a}, url = {http://global-sci.org/intro/article_detail/cicp/11171.html} }
TY - JOUR T1 - A Simple Method for Computing Singular or Nearly Singular Integrals on Closed Surfaces JO - Communications in Computational Physics VL - 3 SP - 733 EP - 753 PY - 2018 DA - 2018/04 SN - 20 DO - http://doi.org/10.4208/cicp.030815.240216a UR - https://global-sci.org/intro/article_detail/cicp/11171.html KW - AB -

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($h^3$), 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.

J. Thomas Beale, Wenjun Ying & Jason R. Wilson. (2020). A Simple Method for Computing Singular or Nearly Singular Integrals on Closed Surfaces. Communications in Computational Physics. 20 (3). 733-753. doi:10.4208/cicp.030815.240216a
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