Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Balancing domain decomposition for problems with large jumps in coefficients


Authors: Jan Mandel and Marian Brezina
Journal: Math. Comp. 65 (1996), 1387-1401
MSC (1991): Primary 65N55, 65F10
DOI: https://doi.org/10.1090/S0025-5718-96-00757-0
MathSciNet review: 1351204
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Balancing Domain Decomposition algorithm uses in each iteration solution of local problems on the subdomains coupled with a coarse problem that is used to propagate the error globally and to guarantee that the possibly singular local problems are consistent. The abstract theory introduced recently by the first-named author is used to develop condition number bounds for conforming linear elements in two and three dimensions. The bounds are independent of arbitrary coefficient jumps between subdomains and of the number of subdomains, and grow only as the squared logarithm of the mesh size $h$. Computational experiments for two- and three-dimensional problems confirm the theory.


References [Enhancements On Off] (What's this?)

  • 1. S. Ashby, T. A. Manteuffel, and P. E. Saylor, A taxonomy for conjugate gradient methods, SIAM J. Numer. Anal., 27 (1990), pp. 1542--1568. MR 91i:65062
  • 2. J.-F. Bourgat, R. Glowinski, P. Le Tallec, and M. Vidrascu, Variational formulation and algorithm for trace operator in domain decomposition calculations, in Domain Decomposition Methods, T. Chan, R. Glowinski, J. Périaux, and O. Widlund, eds., SIAM, Philadelphia, PA, 1989. MR 90b:65198
  • 3. J. H. Bramble, J. E. Pasciak, and A. H. Schatz, The construction of preconditioners for elliptic problems by substructuring, I, Math. Comp., 47 (1986), pp. 103--134. MR 87m:65174
  • 4. ------, An iterative method for elliptic problems on regions partitioned into substructures, Math. Comp., 46 (1986), pp. 361--369. MR 88a:65123
  • 5. ------, The construction of preconditioners for elliptic problems by substructuring, IV, Math. Comp., 53 (1989), pp. 1--24. MR 89m:65098
  • 6. T. F. Chan, Analysis of preconditioners for domain decomposition, SIAM J. Numer. Anal., 24 (1987), pp. 382--390. MR 88e:65033
  • 7. T. F. Chan and T. P. Mathew, The interface probing technique in domain decomposition, SIAM J. on Matrix Analysis and Applications, 13 (1992), pp. 212--238. MR 92i:65183
  • 8. P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North--Holland, Amsterdam, 1978. MR 58:25001
  • 9. L. Cowsar, J. Mandel, and M. F. Wheeler, Balancing domain decomposition for mixed finite elements. Math. Comp., 64 (1995), pp. 989--1015. MR 95j:65161
  • 10. Y.-H. De Roeck, Résolution sur Ordinateurs Multi-Processeurs de Problème d'Elasticité par Décomposition de Domaines, PhD thesis, Université Paris IX Daupine, 1991.
  • 11. Y.-H. De Roeck and P. Le Tallec, Analysis and test of a local domain decomposition preconditioner, in Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski, Y. Kuznetsov, G. Meurant, J. Périaux, and O. Widlund, eds., SIAM, Philadelphia, PA, 1991. CMP 91:12
  • 12. M. Dryja, A method of domain decomposition for 3-D finite element elliptic problems, 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. MR 90b:65200
  • 13. M. Dryja and O. B. Widlund, Towards a unified theory of domain decomposition algorithms for elliptic problems, in Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, held in Houston, Texas, March 20-22, 1989, T. Chan, R. Glowinski, J. Périaux, and O. Widlund, eds., SIAM, Philadelphia, PA, 1990. MR 91m:65294
  • 14. ------, Additive Schwarz methods for elliptic finite element problems in three dimensions, in Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations, T. F. Chan, D. E. Keyes, G. A. Meurant, J. S. Scroggs, and R. G. Voigt, eds., SIAM, Philadelphia, PA, 1992. MR 93j:65201
  • 15. ------, Domain decomposition algorithms with small overlap, SIAM J. Sci.Comput., 15 (1994), pp. 604--620. MR 95d:65102
  • 16. ------, Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems, Comm. Pure Appl. Math, 48 (1995), pp. 121--155. CMP 95:09
  • 17. R. Glowinski and M. F. Wheeler, Domain decomposition and mixed finite element methods for elliptic problems, 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. MR 90a:65237
  • 18. D. E. Keyes and W. D. Gropp, A comparison of domain decomposition techniques for elliptic partial differential equations and their parallel implementation, SIAM J. Sci. Stat. Comput., 8 (1987), pp. s166--s202. MR 88g:65101
  • 19. P. Le Tallec, J. Mandel, and M. Vidrascu, Balancing domain decomposition for plates, in Domain Decomposition Methods in Scientific and Engineering Computing, D. E. Keyes and J. Xu, eds., American Mathematical Society, Providence, RI, 1994, pp. 515--524. Proceedings of the 7th International Symposium on Domain Decomposition Methods, Penn State, November 1993. MR 95j:73080
  • 20. J. Mandel, Balancing domain decomposition, Comm. in Numerical Methods in Engrg., 9 (1993), pp. 233--241. MR 94b:65158
  • 21. J. Mandel and M. Brezina, Balancing domain decomposition: Theory and computations in two and three dimensions, UCD/CCM Report 2, Center for Computational Mathematics, University of Colorado at Denver, November 1993.
  • 22. J. Mandel, S. McCormick, and R. Bank, Variational multigrid theory, in Multigrid Methods, S. F. McCormick, ed., SIAM, Philadephia, 1987, ch. 5, pp. 131--177. CMP 21:05
  • 23. M. Sarkis, Two-level Schwarz methods for nonconforming finite elements and discontinuous coefficients, in Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods, Volume 2, N. D. Melson, T. A. Manteuffel, and S. F. McCormick, eds., no. 3224, Hampton VA, 1993, NASA, pp. 543--566.
  • 24. Scientific Computing Associates, CLAM User's Guide; The Computational Linear Algebra Machine, Version 2.00, New Haven, CT, 1990.
  • 25. B. F. Smith, A domain decomposition algorithm for elliptic problems in three dimensions, Numer. Math., 60 (1991), pp. 219--234. MR 92m:65159
  • 26. O. B. Widlund, An extension theorem for finite element spaces with three applications, in Numerical Techniques in Continuum Mechanics, W. Hackbusch and K. Witsch, eds., Braunschweig/Wiesbaden, 1987, Notes on Numerical Fluid Mechanics, v. 16, Friedr. Vieweg und Sohn, pp. 110--122. Proceedings of the Second GAMM-Seminar, Kiel, January, 1986.
  • 27. ------, Iterative substructuring methods: Algorithms and theory for elliptic problems in the plane, 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. MR 90c:65138

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 65N55, 65F10

Retrieve articles in all journals with MSC (1991): 65N55, 65F10


Additional Information

Jan Mandel
Affiliation: Center for Computational Mathematics, University of Colorado at Denver, Denver, Colorado 80217-3364
Email: jmandel@colorado.edu

Marian Brezina
Affiliation: Center for Computational Mathematics, University of Colorado at Denver, Denver, Colorado 80217-3364
Email: mbrezina@carbon.denver.colorado.edu

DOI: https://doi.org/10.1090/S0025-5718-96-00757-0
Keywords: Domain decomposition, second-order elliptic boundary value problems, two-level iterative methods, discontinuous coefficients
Received by editor(s): March 18, 1993
Received by editor(s) in revised form: December 2, 1993, and September 21, 1994
Additional Notes: Submitted March 1993; revised September 1994.
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society