Dirac delta methods for Helmholtz transmission problems
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Dirac delta methods for Helmholtz transmission problems V. Domínguez · M.-L. Rapún · F.-J. Sayas
Received: 19 October 2004 / Accepted: 15 May 2006 / Published online: 25 November 2006 © Springer Science+Business Media B.V. 2006
Abstract In this paper we use a boundary integral method with single layer potentials to solve a class of Helmholtz transmission problems in the plane. We propose and analyze a novel and very simple quadrature method to solve numerically the equivalent system of integral equations which provides an approximation of the solution of the original problem with linear convergence (quadratic in some special cases). Furthermore, we also investigate a modified quadrature approximation based on the ideas of qualocation methods. This new scheme is again extremely simple to implement and has order three in weak norms. Mathematics Subject Classifications (2000) 65R20 · 65N38 Keywords boundary integral equations · Helmholtz transmission problems · quadrature methods · qualocation · Dirac delta
1. Introduction Traditionally, Helmholtz transmission problem (HTP) is the name given to a system of Helmholtz equations with different wave numbers, one on a bounded domain and the other on its complement, coupled through continuity conditions for the unknown and some related fluxes. A relevant field where these problems appear
V. Domínguez (B) · M.-L. Rapún Departamento Matemática e Informática, Universidad Pública de Navarra, Campus de Arrosadía, 31006 Pamplona, Spain e-mail: [email protected] M.-L. Rapún e-mail: [email protected] F.-J. Sayas Departamento Matemática Aplicada, Universidad de Zaragoza C.P.S., 50018 Zaragoza, Spain e-mail: [email protected]
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Adv Comput Math (2008) 28:119–139
is the scattering of acoustic waves in locally homogeneous media in time-harmonic regime. This has led to extensive analytical and numerical studies, aiming at obtaining reliable simulations and at paving the way for two important related problems: electromagnetic waves and associated inverse problems. The books [8, 9] deal with direct and inverse problems for the Helmholtz equation, with an emphasis on exterior boundary value problems and on scattering in non-absorbing media. Although this kind of problems have led research on the Helmholtz equation on unbounded domains, transmission problems have also received attention in the last decades. Different formulations using boundary integral equations can be found for instance in [10, 13, 15, 26, 27]. More recently, HTP have also appeared in the analysis of the scattering of thermal waves [17, 18], based on related work in physical literature [16, 24, 25]. Also, the use of the Laplace transform with numerical quadrature for the inversion formula on special contours [12] allows for the transformation of evolutionary problems into a set of steady-state Helmholtz equations for several wave numbers. In all these cases, transmission problems are more relevant than purely exterior BVP and the media have absorbtion. In this work we deal with an indirect formulat
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