Niall Bootland,
Victorita Dolean,
Alexander Kyriakis,
Jennifer Pestana
S. 112 - 141
doi:
10.1553/etna_vol55s112
Verlag der Österreichischen Akademie der Wissenschaften
doi:
10.1553/etna_vol55s112
Abstract:
In this work we study the convergence properties of the one-level parallel Schwarz method with Robin transmission conditions applied to the one-dimensional and two-dimensional Helmholtz and Maxwell's equations. One-level methods are not scalable in general. However, it has recently been proven that when impedance transmission conditions are used in the case of the algorithm being applied to the equations with absorption, then, under certain assumptions, scalability can be achieved and no coarse space is required. We show here that this result is also true for the iterative version of the method at the continuous level for strip-wise decompositions into subdomains that are typically encountered when solving wave-guide problems. The convergence proof relies on the particular block Toeplitz structure of the global iteration matrix. Although non-Hermitian, we prove that its limiting spectrum has a near identical form to that of a Hermitian matrix of the same structure. We illustrate our results with numerical experiments.
domain decomposition methods, Helmholtz equations, Maxwell equations, Schwarz algorithms, one-level methods, block Toeplitz matrices
Published Online:
2021/11/09 13:13:01
Object Identifier:
0xc1aa5576 0x003cfd0d
Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613.
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