Journal of Integral Equations and Applications

Well-posed boundary integral equation formulations and Nyström discretizations for the solution of Helmholtz transmission problems in two-dimensional Lipschitz domains

Víctor Domínguez, Mark Lyon, and Catalin Turc

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We present a comparison among the performance of solvers based on Nystr\"om discretizations of several well-posed boundary integral equation formulations of Helmholtz transmission problems in two-dimensional Lipschitz domains. Specifically, we focus on the following four classes of boundary integral formulations of Helmholtz transmission problems: (1)~the classical first kind integral equations for transmission problems~\cite {costabel-stephan}, (2)~the classical second kind integral equations for transmission problems~\cite {KressRoach}, (3)~the single integral equation formulations~\cite {KleinmanMartin}, and (4)~certain direct counterparts of recently introduced generalized combined source integral equations \cite {turc2, turc3}. The former two formulations were the only formulations whose well-posedness in Lipschitz domains was rigorously established \cite {costabel-stephan, ToWe:1993}. We establish the well-posedness of the latter two formulations in appropriate functional spaces of boundary traces of solutions of transmission Helmholtz problems in Lipschitz domains. We give ample numerical evidence that Nystr\"om solvers based on formulations (3) and (4) are computationally more advantageous than solvers based on the classical formulations (1) and (2), especially in the case of high-contrast transmission problems at high frequencies.

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J. Integral Equations Applications, Volume 28, Number 3 (2016), 395-440.

First available in Project Euclid: 17 October 2016

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Primary: 35J05: Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation [See also 31Axx, 31Bxx] 65F08: Preconditioners for iterative methods 65N38: Boundary element methods 65T40: Trigonometric approximation and interpolation

Transmission problems integral equations Lipschitz domains regularizing operators Nyström method graded meshes


Domínguez, Víctor; Lyon, Mark; Turc, Catalin. Well-posed boundary integral equation formulations and Nyström discretizations for the solution of Helmholtz transmission problems in two-dimensional Lipschitz domains. J. Integral Equations Applications 28 (2016), no. 3, 395--440. doi:10.1216/JIE-2016-28-3-395.

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