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.
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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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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