Numerical Algorithms

, Volume 8, Issue 2, pp 329–346 | Cite as

Additive Schwarz domain decomposition methods for elliptic problems on unstructured meshes

  • Tony F. Chan
  • Jun Zou
Article

Abstract

We give several additive Schwarz domain decomposition methods for solving finite element problems which arise from the discretizations of elliptic problems on general unstructured meshes in two and three dimensions. Our theory requires no assumption (for the main results) on the substructures which constitute the whole domain, so each substructure can be of arbitrary shape and of different size. The global coarse mesh is allowed to be non-nested to the fine grid on which the discrete problem is to be solved and both the coarse meshes and the fine meshes need not be quasi-uniform. In this general setting, our algorithms have the same optimal convergence rate of the usual domain decomposition methods on structured meshes. The condition numbers of the preconditioned systems depend only on the (possibly small) overlap of the substructures and the size of the coares grid, but is independent of the sizes of the subdomains.

Keywords

Unstructured meshes non-nested coarse meshes additive Schwarz algorithm optimal convergence rate 

AMS(MOS) subject classification

65N30 65F10 

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References

  1. [1]
    T.J. Barth, Aspects of unstructured grids and finite-volume solvers for the Euler and Navier-Stokes equations, in:Special Course on Unstructured Grid Methods for Advection Dominated Flows, special course at the VKI, Belgium (1992).Google Scholar
  2. [2]
    J. Bramble and J. Xu, Some estimates for a weightedL 2 projection, Math. Comp. 56 (1991) 463–476.Google Scholar
  3. [3]
    J.H. Bramble, J.E. Pasciak, J. Wang and J. Xu, Convergence estimates for product iterative methods with applications to domain decomposition. Math. Comp., 57 (1991) 1–21.Google Scholar
  4. [4]
    X.C. Cai, The use of pointwise interpolation in domain decomposition methods with nonnested meshes, SIAM J. Sci. Comp. 16 (1995).Google Scholar
  5. [5]
    X.C. Cai and Y. Saad, Overlapping domain decomposition algorithms for general sparse matrices, Technical Report Tech. Rep. 93-027, Army High Performance Comp. Res., Center, Univ. of Minn. (March 1993).Google Scholar
  6. [6]
    T. Chan, B. Smith and J. Zou, Multigrid and domain decomposition methods for unstructured meshes, in:Proc. 3rd Int. Conf. on Advances in Numerical Methods and Applications, Sofia, Bulgaria, eds. I.T. Dimov, Bl. Sendov and P. Vassilevski (World Scientific, 1994) pp. 53–62.Google Scholar
  7. [7]
    T. Chan, B. Smith and J. Zou, Overlapping Schwarz methods on unstructured meshes using non-matching coarse grids, Technical Report 94-8, Department of Mathematics, University of California at Los Angeles (February 1994).Google Scholar
  8. [8]
    T.F. Chan and T.P. Mathew, Domain decomposition algorithms, Acta Numerica (1994) 61–143.Google Scholar
  9. [9]
    T.F. Chan and B. Smith, Domain decomposition and multigrid methods for elliptic problems on unstructured meshes, in:Domain Decomposition Methods in Science and Engineering, Proc. 7th Int. Conf. on Domain Decomposition, eds. D. Keyes and J. Xu (American Mathematical Society, Providence, 1995).Google Scholar
  10. [10]
    Z. Chen and J. Zou, An optimal preconditioned GMRES method for general parabolic problems, Report No. 436, DFG-SPP-Anwendungsbezogene Optimierung und Steuerung (1993).Google Scholar
  11. [11]
    P. Ciarlet,The Finite Element Methods for Elliptic Problems (North-Holland, 1978).Google Scholar
  12. [12]
    P. Clément, Approximation by finite element functions using local regularization, RAIRO Numer. Anal. R-2 (1975) 77–84.Google Scholar
  13. [13]
    M. Dryja, B. Smith and O. Widlund, Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions, Technical Report 638, Department of Computer Science, Courant Institute (May 1993), to appear in SIAM J. Number. Anal.Google Scholar
  14. [14]
    M. Dryja and O. Widlund, An additive variant of the Schwarz alternating method for the case of many subregions, Technical Report 339, also Ultracomputer Note 131, Department of Computer Science, Courant Institute (1987).Google Scholar
  15. [15]
    M. Dryja and O. Widlund, Towards a unified theory of domain decomposition algorithms for elliptic problems, in:3rd Int. Symp. on Domain Docomposition Methods for Partial Differential Equations, eds. T. Chan, R. Glowinski, J. Périaux and O. Widlund (SIAM, Philadelphia, PA, 1990).Google Scholar
  16. [16]
    M. Dryja and O. Widlund, Domain decomposition algorithms with small overlap, SIAM J. Sci. Comp. 15 (1994) 604–620.Google Scholar
  17. [17]
    J. Mandel, Hybrid domain decomposition with unstructured subdomains, in:6th Conf. on Domain Decomposition Methods for Partial Differential Equations, ed. A. Quarteroni (AMS, 1993).Google Scholar
  18. [18]
    D.J. Mavriplis, Unstructured mesh algorithms for aerodynamic calculations, Technical Report 92-35, ICASE, NASA Langley, Virginia (July 1992).Google Scholar
  19. [19]
    B. Smith, An optimal domain decomposition preconditioner for the finite element solution of linear elasticity problems, SIAM J. Sci. Stat. Comp. 13 (1992) 364–378.Google Scholar
  20. [20]
    O. Widlund, Iterative substructuring methods: Algorithms and theory for elliptic problems in the plane, in:1st Int. Symp. on Domain Decomposition Methods for Partial Differential Equations, eds. R. Glowinski, G.H. Golub, G.A. Meurant and J. Périaux (SIAM, Philadelphia, PA, 1988).Google Scholar
  21. [21]
    J. Xu, Theory of multilevel methods, PhD thesis, Cornell University (May 1989).Google Scholar
  22. [22]
    J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992) 581–613.Google Scholar

Copyright information

© J.C. Baltzer AG, Science Publishers 1994

Authors and Affiliations

  • Tony F. Chan
    • 1
  • Jun Zou
    • 1
    • 2
  1. 1.Department of MathematicsUniversity of California at Los AngelesLos AngelesUSA
  2. 2.Computing CenterChinese Academy of SciencesBeijingPR China

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