Commun. Comput. Phys., 14 (2013), pp. 1073-1093.


A Fast Accurate Boundary Integral Method for Potentials on Closely Packed Cells

Wenjun Ying 1, J. Thomas Beale 2*

1 Shanghai Jiao Tong University, Department of Mathematics and Institute of Natural Sciences, Minhang, Shanghai 200240, P.R. China.
2 Duke University, Department of Mathematics, Durham, NC 27708-0320, USA.

Received 21 June 2012; Accepted (in revised version) 24 January 2013
Available online 10 May 2013
doi:10.4208/cicp.210612.240113a

Abstract

Boundary integral methods are naturally suited for the computation of harmonic functions on a region having inclusions or cells with different material properties. However, accuracy deteriorates when the cell boundaries are close to each other. We present a boundary integral method in two dimensions which is specially designed to maintain second order accuracy even if boundaries are arbitrarily close. The method uses a regularization of the integral kernel which admits analytically determined corrections to maintain accuracy. For boundaries with many components we use the fast multipole method for efficient summation. We compute electric potentials on a domain with cells whose conductivity differs from that of the surrounding medium. We first solve an integral equation for a source term on the cell interfaces and then find values of the potential near the interfaces via integrals. Finally we use a Poisson solver to extend the potential to a regular grid covering the entire region. A number of examples are presented. We demonstrate that increased refinement is not needed to maintain accuracy as interfaces become very close.

AMS subject classifications: 35J05, 65N06, 65N38

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Key words: Laplace equation, boundary integral method, fast multipole method.

*Corresponding author.
Email: wying@sjtu.edu.cn (W.-J. Ying), beale@math.duke.edu (J. T. Beale)
 

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