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

• Published in: IMA journal of applied mathematics Vol. 74; no. 2; pp. 163 - 177
• Main Authors: Chen, Ke, Cheng, Jin, Harris, Paul J.
• Format: Journal Article
• Language: English
• Published: Oxford Oxford University Press 01.04.2009; Oxford Publishing Limited (England)
• Subjects: boundary integral equation; Burton–Miller; Calculus of variations and optimal control; collocation method; Exact sciences and technology; exterior Helmholtz; Green theorem; hypersingular operators; Integral equations; Mathematical analysis; Mathematics; Operator theory; Ordinary differential equations; Sciences and techniques of general use; collocation method; exterior Helmholtz; boundary integral equation; Green theorem; Burton-Miller; hypersingular operators; Polynomial; Solution uniqueness; Integral operator; Fast solver; Differential equation; Polynomial method; Surface; Kernel function; Applied mathematics; Integral equation; Galerkin method; Collocation method
• ISSN: 0272-4960; 1464-3634
• DOI: 10.1093/imamat/hxp002

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.