Skip to main content
SpringerLink
Log in

Finite element approximation of multi-scale elliptic problems using patches of elements

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

In this paper we present a method for the numerical solution of elliptic problems with multi-scale data using multiple levels of not necessarily nested grids. The method consists in calculating successive corrections to the solution in patches whose discretizations are not necessarily conforming. This paper provides proofs of the results published earlier (see C. R. Acad. Sci. Paris, Ser. I 337 (2003) 679–684), gives a generalization of the latter to more than two domains and contains extensive numerical illustrations. New results including the spectral analysis of the iteration operator and a numerical method to evaluate the constant of the strengthened Cauchy-Buniakowski-Schwarz inequality are presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Achchab, B., Maitre, J.-F.: Estimate of the constant in two strengthened C.B.S. inequalities for F.E.M. systems of 2D elasticiy: Application of multilevel methods and a posteriori error estimators. Numer. Linear Algebra Appl. 3(2), 147–159 (1996)

    Article  Google Scholar 

  2. Achdou, Y., Maday, Y.: The mortar element method with overlapping subdomains. SIAM J. Numer. Anal. 40(2), 601–628 (2002)

    Article  Google Scholar 

  3. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Vol. 140 Pure and Applied Mathematics. Elsevier Science, Oxford, 2nd edition, 2003

  4. Axelsson, O.: On multigrid methods of the two-level type. In: W. Hackbusch, U. Trottenberg (eds.), Multigrid Methods. Vol. 960 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1981 pp. 352–367

  5. Axelsson, O.: Stabilization of algebraic multilevel iteration methods; additive methods. Numer. Algorithms 21, 23–47 (1999)

    Article  Google Scholar 

  6. Axelsson, O., Blaheta, R.: Two simple derivations of universal bounds for the C.B.S. inequality constant. Report No. 0133, Dept. of Mathematics, University of Nijmegen, The Netherlands, 2001

  7. Axelsson, O., Gustafsson, I.: Preconditioning and Two-Level Multigrid methods of arbitrary degree of approximation. Math. Comp. 40(161), 219–242 (1983)

    Google Scholar 

  8. Axelsson, O., Vassilevski, P.S.: Algebraic multilevel preconditioning methods. I. Numer. Math. 56, 157–177 (1989)

    Article  Google Scholar 

  9. Axelsson, O., Vassilevski, P.S.: Algebraic multilevel preconditioning methods. II. SIAM J. Numer. Anal. 27(6), 1569–1590 (1990)

    Article  Google Scholar 

  10. Bank, R.E., Dupont, T.F., Yserentant, H.: The hierarchical basis multigrid method. Numer. Math. 52, 427–458 (1988)

    Article  Google Scholar 

  11. Bank, R.E., Smith, R.K.: A posteriori error estimates based on hierarchical bases. SIAM J. Numer. Anal.30(4), 921–935 (1993)

    Google Scholar 

  12. Blaheta, R.: Space decomposition preconditioners and parallel solvers. In Conference Proceedings ENUMATH 2003, Prague. Springer-Verlag, to appear

  13. Braess, D.: The Contraction Number of a Multigrid Method for Solving the Poisson Equation. Numer. Math. 37, 387–404 (1981)

    Article  Google Scholar 

  14. Braess, D.: The Convergence Rate of a Multigrid Method with Gauss-Seidel Relaxation for the Poisson Equation. In: W. Hackbusch, U. Trottenberg(eds.), Multigrid Methods. Vol. 960 Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1981 pp. 368–386

  15. Bramble, J.H., Ewing, R.E., Pasciak, J.E., Schatz, A.H.: A preconditioning technique for the efficient solution of problems with local grid refinement. Comput. Methods Appl. Mech. Engrg. 67, 149–159 (1988)

    Article  Google Scholar 

  16. Bramble, J.H., Pasciak, J.E., Wang, J., Xu, J.: Convergence estimates for product iterative methods with applications to domain decomposition. Math. Comp. 57(195), 1–21 (1991)

    Google Scholar 

  17. Brezzi, F., Lions, J.-L., Pironneau, O.: Analysis of a Chimera method. C. R. Acad. Sci. Paris Ser. I 332, 655–660 (2003)

    Google Scholar 

  18. Cai, X.-C., Widlund, O.B.: Multiplicative Schwarz algorithms for some nonsymmetric and indefinite problems. SIAM J. Numer. Anal. 30(4), 936–952 (1993)

    Article  Google Scholar 

  19. Chan, T.F., Smith, B.F., Zou, J.: Overlapping Schwarz methods on unstructured meshes using non-matching coarse grids. Numer. Math. 73, 149–167 (1996)

    Article  Google Scholar 

  20. Ciarlet, Ph.G.: Numerical Analysis of the Finite Element Method. Séminaire de Matheématiques Supérieures. Les Presses de L'Université de Montréal, Canada, 1976

  21. Eijkhout, V., Vassilevski, P.: The role of the strengthened Cauchy-Buniakowskii-Schwarz inequality in multilevel methods. SIAM Rev. 33(3), 405–419 (1991)

    Article  Google Scholar 

  22. Glowinski, R., He, J., Rappaz, J., Wagner, J.: Approximation of multi-scale elliptic problems using patches of finite elements. C. R. Acad. Sci. Paris Ser. I 337, 679–684 (2003)

    Google Scholar 

  23. Griebel, M., Oswald, P.: On the abstract theory of additive and multiplicative Schwarz algorithms. Numer. Math. 70, 163–180 (1995)

    Article  Google Scholar 

  24. Hecht, F., Pironneau, O., Ohtsuka, K.: Freefem++ ver. 1.28. http://www.freefem.org

  25. Jung, M., Maitre, J.-F.: Some Remarks on the Constant in the Strengthened CBS Inequality: Estimate for hierarchical finite element discretizations of elasticity problems. Numer. Methods Partial Differential Equations 15(4), 469–487 (1999)

    Article  Google Scholar 

  26. Laydi, M.R.: Convergence d'un schéma itératif explicite. C. R. Acad. Sci. Paris Ser. I 326, 511–514 (1998)

    Google Scholar 

  27. Berlin Maitre, J.-F., Musy, F.: The contraction number of a class of two-level methods; an exact evaluation for some finite element subspaces and model problems. In: W. Hackbusch, U. Trottenberg (eds.), Multigrid Methods. Vol. 960 of Lecture Notes in Mathematics. Springer-Verlag, 1981, pp. 535–544

  28. Margenov, S.D.: Upper bound of the constant in the strengthened C.B.S. inequality for FEM 2D elasticity equations. Numer. Linear Algebra Appl. 1(1) 65–74 (1994)

    Google Scholar 

  29. McCormick, S.F.: Fast Adaptive Composite grid (FAC) Methods: Theory for the variational Case. In: K. Bohner, H.J. Stetter (eds.), Defect Correction Methods. Theory and Applications. Vol.5 of Computing Supplementum, Springer 1984, pp. 115–121

  30. McCormick, S.F., Ruge, J.W.: Unigrid for multigrid simulation. Math. Comp. 41(163), 43–62 (1983)

    Google Scholar 

  31. McCormick, S.F., Thomas, J.: The Fast Adaptive Composite Grid (FAC) Method for Elliptic Equations. Math. Comp. 46(174), 439–456 (1986)

    Google Scholar 

  32. Quarteroni, A., Valli, A.: Domain Decomposition Methods for Partial Differential Equations. Numerical Mathematics and Scientific Computation. Oxford Science Publications, Oxford, 1999

  33. Schwarz, H.A.: Ueber einen Grenzübergang durch alternirendes Verfahren. Vol. 2 of Gesammelte Mathematische Abhandlungen. Chelsea Publishing Company, Bronx, New York, 1972, pp. 133–143

  34. Smith, B., Bjørstad, P., Gropp, W.: Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, Cambridge, 1996

  35. Steger, J.L., Benek, J.A.: On the use of composite grid schemes in computational aerodynamics. Comput. Methods Appl. Mech. Engrg. 64, 301–320 (1987)

    Article  Google Scholar 

  36. Wang, J.: Convergence analysis of the Schwarz algorithm and multilevel decomposition iterative methods II: Nonselfadjoint and indefinite elliptic problems. SIAM J. Numer. Anal. 30(4), 953–970 (1993)

    Article  Google Scholar 

  37. Widlund, O.B.: Domain decomposition methods for elliptic partial differential equations. In: H. Bulgak, C. Zenger (eds.), Error control and adaptativity in scientific computing. Kluwer Academic Publishers, 1999, pp. 325–354

  38. Wohlmuth, B.I.: Discretization methods and iterative solvers based on domain decomposition. Vol. 17 of Lecture Notes in Computational Science and Engineering. Springer, 1991

  39. Xu, J.: Iterative methods by space decomposition and subspace correction. SIAM Rev. 34(4), 581–613 (1992)

    Article  Google Scholar 

  40. Xu, J.: An introduction to multilevel methods. In: M. Ainsworth, J. Levesley, W.A. Light, M. Marletta (eds.), Wavelets, multilevel methods and elliptic PDEs, Numerical Mathematics and Scientific Computation. Oxford University Press, 1997, pp. 213–302

  41. Xu, J.: The method of subspace corrections. J. Comput. Appl. Math. 128, 335–362 (2001)

    Article  Google Scholar 

  42. Xu, J., Zikatanov, L.: The method of alternating projections and the method of subspace corrections in Hilbert space. PennState, Department of Mathematics, AM223, 2000

  43. Yosida, K.: Functional Analysis. Vol. 123 of A series of Comprehensive Studies in Mathematics. Springer-Verlag, Berlin, 6th edititon, 1980

  44. Yserentant, H.: On the multi-level splitting of finite element spaces. Numer. Math. 49, 379–412 (1986)

    Article  Google Scholar 

  45. Yserentant, H.: Hierarchical Bases. In: R.E. O'Malley (ed), Proceedings of the Second International Conference on Industrial and Applied Mathematics. SIAM, 1991, pp. 256–276

  46. Yserentant, H.: Old and new convergence proofs for multigrid methods. Acta Numer. 285–326 (1993)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Houston, 4800 Calhoun Road, Houston, Texas, 77204-3008, USA

    Roland Glowinski & Jiwen He

  2. Section of Mathematics, Swiss Federal Institute of Technology, 1015, Lausanne, Switzerland

    Alexei Lozinski, Jacques Rappaz & Joël Wagner

Authors
  1. Roland Glowinski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiwen He
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alexei Lozinski
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jacques Rappaz
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Joël Wagner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joël Wagner.

Additional information

Supported by CTI Project 6437.1 IWS-IW.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Glowinski, R., He, J., Lozinski, A. et al. Finite element approximation of multi-scale elliptic problems using patches of elements. Numer. Math. 101, 663–687 (2005). https://doi.org/10.1007/s00211-005-0614-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-005-0614-5

Mathematics Subject Classifications (1991)

Search

Navigation