In this paper we generalize the Aitken-like acceleration method of the additive Schwarz algorithm for elliptic problems to the additive Schwarz waveform relaxation for the heat equation. The domain decomposition is in space and time. The standard Schwarz waveform relaxation algorithm has a linear rate of convergence and low numerical efficiency. This algorithm is, however, friendly to cache use and scales with the memory in parallel environments. We show that our new acceleration procedure of the waveform acceleration algorithm results in a fast direct solver.
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J. Baranger, M. Garbey, and F. Oudin-Dardun. On Aitken like acceleration of Schwarz domain decomposition method using generalized Fourier. Domain Decomposition Methods in Science and Engineering, pages 341 – 348, 2003.
N. Barberou, M. Garbey, M. Hess, M. Resch, T. Rossi, J. Toivanen, and D. Tromeur-Dervout. On the efficient meta-computing of linear and nonlinear elliptic problems. J. Parallel Distrib. Comput. (special issue on grid computing), 63:564 – 577, 2003.
B. Després. Décomposition de domaine et problème de Helmholtz. C. R. Acad. Sci. Paris Sér. I Math., 311(6):313–316, 1990.
M. Gander and H. Zhao. Overlapping Schwarz wavefrom relaxation for the heat equation in n-dimensions. BIT, 40(4):001 – 004, 2000.
M. Garbey. Acceleration of a Schwarz waveform relaxation method for parabolic problems. Preprint UH-CS-06-11. Submitted.
M. Garbey. Acceleration of the Schwarz method for elliptic problem. SIAM J. Sci. Comput., 26(6):1871 – 1893, 2005.
M. Garbey and D. T. Dervout. On some Aitken-like acceleration of the Schwarz method. Internat. J. Numer. Methods Fluids, 40(12):1493 – 1513, 2002.
C. Japhet, F. Nataf, and F. Rogier. The optimized order 2 method. Application to convection-diffusion problems. Future Generation Computer Systems FUTURE, 18, 2001.
V. Martin. An optimized Schwarz waveform relaxation method for unsteady convection-diffusion equation. Appl. Numer. Math., 52(4):401 – 428, 2005.
F. Nataf and F. Rogier. Factorization of the convection-diffusion operator and a (possibly) non overlapping Schwarz method. M3AS, 5(1):67 – 93, 1995.
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Garbey, M. (2008). A Direct Solver for the Heat Equation with Domain Decomposition in Space and Time. In: Langer, U., Discacciati, M., Keyes, D.E., Widlund, O.B., Zulehner, W. (eds) Domain Decomposition Methods in Science and Engineering XVII. Lecture Notes in Computational Science and Engineering, vol 60. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75199-1_63
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DOI: https://doi.org/10.1007/978-3-540-75199-1_63
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