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Optimized Schwarz Methods for Curl-Curl Time-Harmonic Maxwell’s Equations

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Domain Decomposition Methods in Science and Engineering XXI

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 98))

Abstract

Like the Helmholtz equation, the high frequency time-harmonic Maxwell’s equations are difficult to solve by classical iterative methods. Domain decomposition methods are currently most promising: following the first provably convergent method in [4], various optimized Schwarz methods were developed over the last decade [1–3, 5, 8, 10, 11, 13, 14, 16]. There are however two basic formulations for Maxwell’s equation: the first order formulation, for which complete optimized results are known [5], and the second order, or curl-curl formulation, with partial optimization results [1, 13, 16]. We show in this paper that the convergence factors and the optimization process for the two formulations are the same. We then show by numerical experiments that the Fourier analysis predicts very well the behavior of the algorithms for a Yee scheme discretization, which corresponds to Nedelec edge elements on a tensor product mesh, in the curl-curl formulation. When using however mixed type Nedelec elements on an irregular tetrahedral mesh, numerical experiments indicate that transverse magnetic (TM) modes are less well resolved for high frequencies than transverse electric (TE) modes, and a heuristic can then be used to compensate for this in the optimization.

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Author information

Authors and Affiliations

  1. University of Nice-Sophia Antipolis, Nice, France

    Victorita Dolean

  2. University of Geneva, Geneva, Switzerland

    Victorita Dolean & Martin J. Gander

  3. Inria Sophia Antipolis-Méditerranée, Valbonne, France

    Stéphane Lanteri

  4. ElectroScience Laboratory, The Ohio State University, Columbus, OH, USA

    Jin-Fa Lee & Zhen Peng

Authors
  1. Victorita Dolean
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  2. Martin J. Gander
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  3. Stéphane Lanteri
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  4. Jin-Fa Lee
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  5. Zhen Peng
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Corresponding author

Correspondence to Victorita Dolean .

Editor information

Editors and Affiliations

  1. INRIA, Rennes, France

    Jocelyne Erhel

  2. Section de Mathématiques, Université de Genève, Genève, Switzerland

    Martin J. Gander

  3. Laboratoire Analyse, Géometrie & Applications, Université Paris XIII, Villetaneuse, France

    Laurence Halpern

  4. INRIA, Rennes, France

    Géraldine Pichot

  5. Laboratoire LMNO, Université de Caen, Caen, France

    Taoufik Sassi

  6. New York University Courant Institute, New York, New York, USA

    Olof Widlund

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Dolean, V., Gander, M.J., Lanteri, S., Lee, JF., Peng, Z. (2014). Optimized Schwarz Methods for Curl-Curl Time-Harmonic Maxwell’s Equations. In: Erhel, J., Gander, M., Halpern, L., Pichot, G., Sassi, T., Widlund, O. (eds) Domain Decomposition Methods in Science and Engineering XXI. Lecture Notes in Computational Science and Engineering, vol 98. Springer, Cham. https://doi.org/10.1007/978-3-319-05789-7_56

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