Overlapping Schwarz Waveform Relaxation for the Heat Equation in N Dimensions
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We analyze overlapping Schwarz waveform relaxation for the heat equation in n spatial dimensions. We prove linear convergence of the algorithm on unbounded time intervals and superlinear convergence on bounded time intervals. In both cases the convergence rates are shown to depend on the size of the overlap. The linear convergence result depends also on the number of subdomains because it is limited by the classical steady state result of overlapping Schwarz for elliptic problems. However the superlinear convergence result is independent of the number of subdomains. Thus overlapping Schwarz waveform relaxation does not need a coarse space for robust convergence independent of the number of subdomains, if the algorithm is in the superlinear convergence regime. Numerical experiments confirm our analysis. We also briefly describe how our results can be extended to more general parabolic problems.
- M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, U.S. Govt. Print. Off., Washington, DC, 1964.
- A. Bellen and M. Zennaro, The use of Runge-Kutta formulae in waveform relaxation methods, Appl. Numer. Math., 11 (1993), pp. 95-114.
- M. Bjorhus, A note on the convergence of discretized dynamic iteration, BIT, 35 (1995), pp. 291-296.
- K. Burrage, Parallel and Sequential Methods for Ordinary Differential Equations, Oxford University Press, New York, 1995.
- K. Burrage, Z. Jackiewitz, S. P. Norsett, and R. A. Renaut, Preconditioning waveform relaxation iterations for differential systems, BIT, 36 (1996), pp. 54-76.
- X.-C. Cai, Additive Schwarz algorithms for parabolic convection-diffusion equations, Numer. Math., 60 (1991), pp. 41-61.
- X.-C. Cai, Multiplicative Schwarz methods for parabolic problems, SIAMJ. Sci. Comput., 15 (1994), pp. 587-603.
- J. R. Cannon, The One-Dimensional Heat Equation, in Encyclopedia of Mathematics and its Applications, Addison-Wesley, Reading, MA, 1984.
- T. F. Chan and T. P. Mathew, Domain decomposition algorithms, in Acta Numerica 1994, Cambridge University Press, 1994, pp. 61-143.
- A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, 1964.
- M. J. Gander, Overlapping Schwarz for parabolic problems, in Ninth International Conference on Domain Decomposition Methods, University of Bergen, Norway, P. E. Bjorstad, M. Espedal, and D. Keyes, eds., 1997, pp. 97-104.
- M. J. Gander, Overlapping Schwarz waveform relaxation for parabolic problems, in Domain Decomposition Methods 10, The Tenth International Conference on Domain Decomposition Methods, 1997, Boulder, CO, J. Mandel, Ch. Farhat, and X.-C. Cai, eds., Contemporary Mathematics, Vol. 218, AMS, Providence, RI, 1998, pp. 425-431.
- M. J. Gander, A waveform relaxation algorithm with overlapping splitting for reaction diffusion equations, Numer. Linear Algebra Appl., 6 (1998), pp. 125-145.
- M. J. Gander and A. M. Stuart, Space-time continuous analysis of waveform relaxation for the heat equation, SIAM J. Sci. Comput., 19 (1998), pp. 2014-2031.
- M. J. Gander and H. Zhao, Overlapping Schwarz waveform relaxation for parabolic problems in higher dimension, in Proceedings of Algoritmy 14, A. Handlovi?ová, M. Komorníkova, and K. Mikula, eds., Slovak Technical University, September 1997, pp. 42-51.
- E. Giladi and H. Keller, Space time domain decomposition for parabolic problems, Tech. Rep. 97-4, Center for Research on Parallel Computation, California Institute of Technology, Pasadena, CA, 1997.
- J. Janssen and S. Vandewalle, Multigrid waveform relaxation on spatial finite element meshes: The continuous-time case, SIAM J. Numer. Anal., 33 (1996), pp. 456-474.
- J. Janssen and S. Vandewalle, Multigrid waveform relaxation on spatial finite element meshes: The discrete-time case, SIAM J. Sci. Comput., 17 (1996), pp. 133-155.
- R. Jeltschand B. Pohl, Waveform relaxation with overlapping splittings, SIAM J. Sci. Comput., 16 (1995), pp. 40-49.
- G. M. Liberman, Second Order Parabolic Differential Equations, World Scientific Publishing, Singapore, 1996.
- P.-L. Lions, On the Schwarz alternating method. I, in First International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski, G. H. Golub, G. A. Meurant, and J. Périaux, eds., SIAM, Philadelphia, PA, 1988, pp. 1-42.
- C. Lubich and A. Ostermann, Multi-grid dynamic iteration for parabolic equations, BIT, 27 (1987), pp. 216-234.
- U. Miekkala and O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems, SIAM J. Sci. Stat. Comput., 8 (1987), pp. 459-482.
- O. Nevanlinna, Remarks on Picard-Lindelöf iterations. Part I, BIT, 29 (1989), pp. 328-346.
- O. Nevanlinna, Remarks on Picard-Lindelöf iterations. Part II, BIT, 29 (1989), pp. 535-562.
- S. Vandewalle and G. Horton, Fourier mode analysis of the multigrid waveform relaxation and time-parallel multigrid methods, Computing, 54 (1995), pp. 317-330.
- Overlapping Schwarz Waveform Relaxation for the Heat Equation in N Dimensions
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Volume 42, Issue 4 , pp 779-795
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