Mathematics of Computation

Author(s): Serge Kräutle.
Journal: Math. Comp. 74 (2005), 1231-1256.
MSC (2000): Primary 65N55; Secondary 65Y05, 65M70, 35J05
Posted: September 17, 2004
Retrieve article in: PDF

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.

1.: S. Blazy, W. Borchers, U. Dralle, Parallelization Methods for a Characteristic's Pressure Correction Scheme, in: Flow simulation with high-performance computers II, Hirschel (ed.), Notes on Numerical Fluid Dynamics, Vol. 38, Braunschweig, Vieweg 1996. MR 2000g:76001
2.: W. Borchers, M.Y. Forestier, S. Kräutle, R. Pasquetti, R. Peyret, R. Rautmann, N. Roß, C. Sabbah, A Parallel Hybrid Highly Accurate Elliptic Solver for Viscous Flow Problems, Numerical Flow Simulation I, Notes on Num. Fluid Mech. Vol. 66, Hirschel (ed.), pp. 3-24, Springer Verlag 1998.MR 99i:76104
3.: 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.
4.: J. Bramble, J. Pasciak, J. Xu, Parallel multilevel preconditioners, Math. Comp., 55 (1990), pp. 1-22. MR1023042 (90k:65170)
5.: S.C. Brenner, The condition number of the Schur complement in domain decomposition, Numer. Math. 83 (1999), pp. 187-203.MR 2000g:65114
6.: S.C. Brenner, Lower bounds for nonoverlapping domain decomposition preconditioners in two dimensions, Math. Comp. 69 (2000), pp. 1319-1339.MR 2001a:65156
7.: C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dynamics, Springer Verlag, New York 1988. MR 89m:76004
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).
9.: C. Farhat, J. Mandel, F.-X. Roux, Optimal convergence properties of the FETI domain decomposition method, Comput. Methods Appl. Mech. Engrg. 115 (1994), pp. 365-385. MR 95d:65091
10.: 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.
11.: 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.
12.: A. Klawonn, O. Widlund, FETI and Neumann-Neumann iterative substructuring methods: Connections and new results, Comm. Pure Appl. Math. 54 (2001), pp. 57-90.MR 2001i:65131
13.: 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.
14.: J.L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications I, Springer Verlag, Berlin-Heidelberg-New York 1982. MR 50:2670
15.: J.L. Lions, O. Pironneau, Non-overlapping domain decomposition for evolution operators, C. R. Acad. Sci. Paris, t. 330, Série I (2000), pp. 1-8. MR 2001f:65110b
16.: P.L. Lions, On the Schwarz alternating method III: A variant for nonoverlapping subdomains, Proceedings of the Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, Chan, Glowinski, Périaux, Widlund (eds.), Philadelphia, PA, 1990, SIAM, pp. 202-223. MR 91g:65226
17.: J. Mandel, R. Tezaur, Convergence of a substructuring method with Lagrange multipliers, Num. Math. 73 (1996), pp. 473-487.MR 97h:65142
18.: J. Mandel, R. Tezaur, On the convergence of a dual-primal substructuring method, Num. Math. 88 (2001), pp. 543-558. MR 2002b:65161
19.: H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Publishing Company, Amsterdam, New York, Oxford, 1978.MR 80i:46032b

Retrieve articles in Mathematics of Computation with MSC (2000): 65N55, 65Y05, 65M70, 35J05

Serge Kräutle
Affiliation: Institut für Angewandte Mathematik, Universität Erlangen-Nürnberg, Martensstrasse 3, 91054 Erlangen, Germany
Email: kraeutle@am.uni-erlangen.de

DOI: 10.1090/S0025-5718-04-01706-5
PII: S 0025-5718(04)01706-5
Keywords: Parallelization, domain decomposition, preconditioning, FETI
Received by editor(s): October 18, 2003
Received by editor(s) in revised form: February 9, 2004
Posted: September 17, 2004
Additional Notes: This work was supported by the {\em Deutsche Forschungsgemeinschaft} (DFG) and the {\em Centre National de la Recherche Scientifique} (CNRS)
Copyright of article: Copyright 2004, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS