A new study of the Burton and Miller method for the solution of a 3D Helmholtz problem
Chen, K., Cheng, J. and Harris, P.J. (2009) A new study of the Burton and Miller method for the solution of a 3D Helmholtz problem IMA Journal of Applied Mathematics, 74 (2). pp. 163-177. ISSN 0272-4960
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Official URL: http://imamat.oxfordjournals.org/content/74/2/163....
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.
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