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IMA Journal of Applied Mathematics Advance Access originally published online on February 24, 2009
IMA Journal of Applied Mathematics 2009 74(2):163-177; doi:10.1093/imamat/hxp002
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© The Author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

A new study of the Burton and Miller method for the solution of a 3D Helmholtz problem

Ke Chen{dagger}

Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 7ZL, UK

Jin Cheng{ddagger}

Department of Mathematics, Fudan University, Shanghai 200433, China

Paul J. Harris§

School of Computing, Mathematical and Information Sciences, University of Brighton, Lewes Road, Brighton, East Sussex BN2 4GJ, UK

{dagger} Email: k.chen{at}liverpool.ac.uk

{ddagger} Email: jcheng{at}fudan.edu.cn

§ Corresponding author. Email: p.j.harris{at}bton.ac.uk

Received on June 24, 2005; Accepted on December 16, 2008

The exterior Helmholtz problem can be efficiently solved by reformulating the differential equation as an integral equation over the surface of the radiating and/or scattering object. One popular approach for overcoming either non-unique or non-existent problems which occur at certain values of the wave number is the so-called Burton and Miller method which modifies the usual integral equation into one which can be shown to have a unique solution for all real and positive wave numbers. This formulation contains an integral operator with a hypersingular kernel function and for many years, a commonly used method for overcoming this hypersingularity problem has been the collocation method with piecewise-constant polynomials. Viable high-order methods only exist for the more expensive Galerkin method. This paper proposes a new reformulation of the Burton–Miller approach and enables the more practical collocation method to be applied with any high-order piecewise polynomials. This work is expected to lead to much progress in subsequent development of fast solvers. Numerical experiments on 3D domains are included to support the proposed high-order collocation method.

Keywords: exterior Helmholtz; boundary integral equation; Burton–Miller; Green theorem; hypersingular operators; collocation method.


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