A domain decomposition method using efficient interface-acting preconditioners

Kräutle, Serge

doi:10.1090/S0025-5718-04-01706-5

A domain decomposition method using efficient interface-acting preconditioners
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by Serge Kräutle;

Math. Comp. 74 (2005), 1231-1256

DOI: https://doi.org/10.1090/S0025-5718-04-01706-5

Published electronically: September 17, 2004

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Abstract:

The conjugate gradient boundary iteration (CGBI) is a domain decomposition method for symmetric elliptic problems on domains with large aspect ratio. High efficiency is reached by the construction of preconditioners that are acting only on the subdomain interfaces. The theoretical derivation of the method and some numerical results revealing a convergence rate of 0.04–0.1 per iteration step are given in this article. For the solution of the local subdomain problems, both finite element (FE) and spectral Chebyshev methods are considered.

References

Ernst Heinrich Hirschel (ed.), Flow simulation with high-performance computers. II, Notes on Numerical Fluid Mechanics, vol. 52, Friedr. Vieweg & Sohn, Braunschweig, 1996. DFG Priority Research Programme results 1993–1995. MR 1727107, DOI 10.1007/978-3-322-89849-4
W. Borchers, M. Y. Forestier, S. Kräutle, R. Pasquetti, R. Peyret, R. Rautmann, N. Roß, and C. Sabbah, A parallel hybrid highly accurate elliptic solver for viscous flow problems, Numerical flow simulation, I (Marseille, 1997) Notes Numer. Fluid Mech., vol. 66, Friedr. Vieweg, Braunschweig, 1998, pp. 3–24. MR 1668776, DOI 10.1007/978-3-663-10916-7_{1}
W. Borchers, S. Kräutle, R. Pasquetti, R. Peyret, R. Rautmann, Multi-domain Finite Element—Spectral Chebyshev Parallel Navier-Stokes Solver for Viscous Flow Problems, Numerical Flow Simulation III, Notes on Num. Fluid Mech. Vol. 82, Hirschel (ed.), pp. 3–17, Springer Verlag 2003.
J. Bramble, J. Pasciak, J. Xu, Parallel multilevel preconditioners, Math. Comp., 55 (1990), pp. 1–22. MR1023042 (90k:65170)
Susanne C. Brenner, The condition number of the Schur complement in domain decomposition, Numer. Math. 83 (1999), no. 2, 187–203. MR 1712684, DOI 10.1007/s002110050446
Susanne C. Brenner and Li-Yeng Sung, Lower bounds for nonoverlapping domain decomposition preconditioners in two dimensions, Math. Comp. 69 (2000), no. 232, 1319–1339. MR 1710656, DOI 10.1090/S0025-5718-00-01236-9
Claudio Canuto, M. Yousuff Hussaini, Alfio Quarteroni, and Thomas A. Zang, Spectral methods in fluid dynamics, Springer Series in Computational Physics, Springer-Verlag, New York, 1988. MR 917480, DOI 10.1007/978-3-642-84108-8
C. Farhat, F.-X. Roux, A method of finite element tearing and interconnecting and its parallel solution algorithm, Int. J. Num. Methods Engin. 32, pp. 1205–1227 (1991).
Charbel Farhat, Jan Mandel, and François-Xavier Roux, Optimal convergence properties of the FETI domain decomposition method, Comput. Methods Appl. Mech. Engrg. 115 (1994), no. 3-4, 365–385. MR 1285024, DOI 10.1016/0045-7825(94)90068-X
C. Farhat, L. Crivelli, F.-X. Roux, A transient FETI methodology for large-scale parallel implicit computations in structural mechanics, Int. J. Numer. Methods Eng. 37 (1994), pp. 1945-1975.
W. Guo, L.S. Hou, Generalizations and accelerations of Lions’ nonoverlapping domain decomposition method for linear elliptic PDE, SIAM J. Numer. Anal. 41 (2003), pp. 2056–2080.
Axel Klawonn and Olof B. Widlund, FETI and Neumann-Neumann iterative substructuring methods: connections and new results, Comm. Pure Appl. Math. 54 (2001), no. 1, 57–90. MR 1787107, DOI 10.1002/1097-0312(200101)54:1<57::AID-CPA3>3.0.CO;2-D
S. Kräutle, A Navier-Stokes solver based on CGBI and the method of characteristics, Doctoral thesis, Erlangen, 2001. http://www.am.uni-erlangen.de/am1/publications/dipl_phd_thesis/ PhD_Kraeutle.ps.gz.
J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth. MR 350177
Jacques-Louis Lions and Olivier Pironneau, Overlapping domain decomposition of evolution operators, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 10, 937–942 (English, with English and French summaries). MR 1771962, DOI 10.1016/S0764-4442(00)00295-0
P.-L. Lions, On the Schwarz alternating method. III. A variant for nonoverlapping subdomains, Third International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX, 1989) SIAM, Philadelphia, PA, 1990, pp. 202–223. MR 1064345
Jan Mandel and Radek Tezaur, Convergence of a substructuring method with Lagrange multipliers, Numer. Math. 73 (1996), no. 4, 473–487. MR 1393176, DOI 10.1007/s002110050201
Jan Mandel and Radek Tezaur, On the convergence of a dual-primal substructuring method, Numer. Math. 88 (2001), no. 3, 543–558. MR 1835470, DOI 10.1007/s211-001-8014-1
H. Triebel, Interpolation theory, function spaces, differential operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. MR 500580