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Robin-Dirichlet algorithms for the Cauchy problem for the Helmholtz equation
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0002-2681-8965
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Univ Rwanda, Rwanda.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
2018 (English)In: Inverse Problems in Science and Engineering, ISSN 1741-5977, E-ISSN 1741-5985, Vol. 26, no 7, p. 1062-1078Article in journal (Refereed) Published
Abstract [en]

The Cauchy problem for the Helmholtz equation is considered. It was demonstrated in a previous paper by the authors that the alternating algorithm suggested by V.A. Kozlov and V.G. Mazya does not converge for large wavenumbers k in the Helmholtz equation. Here, we present some simple modifications of the algorithm which may restore the convergence. They consist of the replacement of the Neumann-Dirichlet iterations by the Robin-Dirichlet ones which repairs the convergence for less than the first Dirichlet-Laplacian eigenvalue. In order to treat large wavenumbers, we present an algorithm based on iterative solution of Robin-Dirichlet boundary value problems in a sufficiently narrow border strip. Numerical implementations obtained using the finite difference method are presented. The numerical results illustrate that the algorithms suggested in this paper, produce convergent iterative sequences.

Place, publisher, year, edition, pages
TAYLOR & FRANCIS LTD , 2018. Vol. 26, no 7, p. 1062-1078
Keywords [en]
Helmholtz equation; Cauchy problem; alternating iterative method; inverse problem; ill-posed problem
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-147448DOI: 10.1080/17415977.2017.1380639ISI: 000429229600007OAI: oai:DiVA.org:liu-147448DiVA, id: diva2:1206326
Note

Funding Agencies|Swedish International Development Cooperation Agency (Sida); University of Rwanda (UR) [51160027-02, 51160059-02]

Available from: 2018-05-16 Created: 2018-05-16 Last updated: 2018-05-16

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Berntsson, FredrikKozlov, VladimirTuresson, Bengt-Ove
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