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Networks and Heterogeneous Media

2010, Volume 5, Issue 3: 487-505. doi: 10.3934/nhm.2010.5.487
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Schwarz waveform relaxation algorithms for semilinear reaction-diffusion equations

  • 1.
    Univ. Paris-Sud, Département de Mathématiques, CNRS, F-91405 Orsay
  • 2.
    Section de Mathématiques, Université de Genève, CP 64, 1211 Genève
  • 3.
    Université Paris 13, CNRS, UMR 7539 LAGA, 99 av. Jean-Baptiste Clément, F-93430 Villetaneuse
  • 4.
    DMA, Ecole Normale Supérieure, 45 rue d’Ulm, Paris
  • Received: 01 January 2010 Revised: 01 May 2010

  • 35K55, 65M12, 65M55.

  • We introduce nonoverlapping domain decomposition algorithms of Schwarz waveform relaxation type for the semilinear reaction-diffusion equation. We define linear Robin and second order (or Ventcell) transmission conditions between the subdomains, which we prove to lead to a well defined and converging algorithm. We also propose nonlinear transmission conditions. Both types are based on best approximation problems for the linear equation and provide efficient algorithms, as the numerical results that we present here show.

    Citation: Filipa Caetano, Martin J. Gander, Laurence Halpern, Jérémie Szeftel. Schwarz waveform relaxation algorithms for semilinear reaction-diffusion equations[J]. Networks and Heterogeneous Media, 2010, 5(3): 487-505. doi: 10.3934/nhm.2010.5.487

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  • We introduce nonoverlapping domain decomposition algorithms of Schwarz waveform relaxation type for the semilinear reaction-diffusion equation. We define linear Robin and second order (or Ventcell) transmission conditions between the subdomains, which we prove to lead to a well defined and converging algorithm. We also propose nonlinear transmission conditions. Both types are based on best approximation problems for the linear equation and provide efficient algorithms, as the numerical results that we present here show.


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