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A Time-Dependent Dirichlet-Neumann Method for the Heat Equation

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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

We present a waveform relaxation version of the Dirichlet-Neumann method for parabolic problem. Like the Dirichlet-Neumann method for steady problems, the method is based on a non-overlapping spatial domain decomposition, and the iteration involves subdomain solves with Dirichlet boundary conditions followed by subdomain solves with Neumann boundary conditions. However, each subdomain problem is now in space and time, and the interface conditions are also time-dependent. Using a Laplace transform argument, we show for the heat equation that when we consider finite time intervals, the Dirichlet-Neumann method converges, similar to the case of Schwarz waveform relaxation algorithms. The convergence rate depends on the length of the subdomains as well as the size of the time window. In this discussion, we only stick to the linear bound. We illustrate our results with numerical experiments.

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Notes

  1. 1.

    Assuming \(s = re^{i\vartheta }\), \(z = \sqrt{s}\), we can write for b ≥ a, \(\big\vert s^{p}G(s)\big\vert \leq \big\vert \frac{s^{p}} {\cosh (az)}\big\vert \leq \frac{2r^{p}} {\vert e^{a\sqrt{r/2}}-e^{-a\sqrt{r/2}}\vert } \rightarrow 0\), as r → ∞; and for a > b, \(\big\vert s^{p}G(s)\big\vert \leq \big\vert \frac{s^{p}} {\sinh (bz)}\big\vert \leq \frac{2r^{p}} {\vert e^{b\sqrt{r/2}}-e^{-b\sqrt{r/2}}\vert } \rightarrow 0\), as r → ∞.

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Acknowledgements

I would like to express my gratitude to Prof. Martin J. Gander and Dr. Felix Kwok for their constant support and stimulating suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University of Geneva, Geneva, Switzerland

    Bankim C. Mandal

Authors
  1. Bankim C. Mandal
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Correspondence to Bankim C. Mandal .

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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Mandal, B.C. (2014). A Time-Dependent Dirichlet-Neumann Method for the Heat Equation. 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_44

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