Numerische Mathematik

, Volume 104, Issue 3, pp 271–296 | Cite as

Positive definite balancing Neumann–Neumann preconditioners for nearly incompressible elasticity

  • L. Beirão da Veiga
  • C. Lovadina
  • L. F. Pavarino
Article
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Abstract

In this paper, a positive definite Balancing Neumann–Neumann (BNN) solver for the linear elasticity system is constructed and analyzed. The solver implicitly eliminates the interior degrees of freedom in each subdomain and solves iteratively the resulting Schur complement, involving only interface displacements, using a BNN preconditioner based on the solution of a coarse elasticity problem and local elasticity problems with natural and essential boundary conditions. While the Schur complement becomes increasingly ill-conditioned as the materials becomes almost incompressible, the BNN preconditioned operator remains well conditioned. The main theoretical result of the paper shows that the proposed BNN method is scalable and quasi-optimal in the constant coefficient case. This bound holds for material parameters arbitrarily close to the incompressible limit. While this result is due to an underlying mixed formulation of the problem, both the interface problem and the preconditioner are positive definite. Numerical results in two and three dimensions confirm these good convergence properties and the robustness of the methods with respect to the almost incompressibility of the material.

AMS Subject Classifications

65F10 65N30 65N55 
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References

  1. 1.
    Achdou Y., Le Tallec P., Nataf F., Vidrascu M. (2000): A domain decomposition preconditioner for an advection-diffusion problem. Comput. Methods Appl. Mech. Eng. 184, 145–170MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ainsworth M., Sherwin S. (1999): Domain decomposition preconditioners for p and hp finite element approximation of Stokes equations. Comput. Methods Appl. Mech. Eng. 175(3–4): 243–266MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alart P., Barboteu M., Le Tallec P., Vidrascu M. (2000): Méthode de Schwarz additive avec solveur grossier pour problèmes non symétriques. Comp. Rend. Acad. Sci. Paris Ser. I 331: 399–404MATHMathSciNetGoogle Scholar
  4. 4.
    Auricchio F., Brezzi F., Lovadina C.(2004). Mixed finite element methods. In: Stein E., de Borst R., Hughes T.M.J. (eds). Encyclopedia of Computational Mechanics. vol. 1, chap. 9. Wiley, New YorkGoogle Scholar
  5. 5.
    Bernardi C., Maday Y. (1999): Uniform inf–sup conditions for the spectral discretization of the Stokes problem. Math. Models Methods Appl. Sci. 9(3): 395–414MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bramble J.H., Pasciak J.E. (1989/1990): A domain decomposition technique for Stokes problems. Appl. Numer. Math. 6, 251–261CrossRefMathSciNetGoogle Scholar
  7. 7.
    Brezzi F., Fortin M. (1991): Mixed and Hybrid Finite Element Methods. Springer, Berlin Heidelberg New YorkMATHGoogle Scholar
  8. 8.
    Casarin, M.A.: Schwarz Preconditioners for Spectral and Mortar Finite Element Methods with Applications to Incompressible Fluids. PhD thesis, Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, March 1996. Technical Report 717, Department of Computer Science, Courant InstituteGoogle Scholar
  9. 9.
    Cowsar L.C., Mandel J., Wheeler M.F. (1995): Balancing domain decomposition for mixed finite elements. Math. Comput. 64(211): 989–1015MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Dohrmann C.R. (2003): A preconditioner for substructuring based on constrained energy minimization. SIAM J. Sci. Comput. 25(1): 246–258MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dryja M., Widlund O.B. (1995): Schwarz methods of Neumann–Neumann type for three-dimensional elliptic finite element problems. Commun. Pure Appl. Math. 48(2), 121–155MathSciNetGoogle Scholar
  12. 12.
    Farhat C., Lesoinne M., Le Tallec P., Pierson K., Rixen D. (2001): FETI-DP: a dual-primal unified FETI method - part I: a faster alternative to the two-level FETI method. Int. J. Numer. Methods. Eng. 50, 1523–1544MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Fischer P.F. (1997): An overlapping Schwarz method for spectral element solution of the incompressible Navier–Stokes equations. J. Comput. Phys. 133(1): 84–101MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Fischer P.F., Miller N.I., Tufo H.M. (2000): An overlapping Schwarz method for spectral element simulation of three-dimensional incompressible flows. In: Bjørstad P., Luskin M. (eds). Parallel Solution of PDE. Mathematics and Its Applications, vol 120. Springer, Berlin Heidelberg New York, pp. 1–30Google Scholar
  15. 15.
    Fischer P.F., Rønquist E. (1994): Spectral element methods for large scale parallel Navier–Stokes calculations. Comput. Methods Appl. Mech. Eng. 116, 69–76MATHCrossRefGoogle Scholar
  16. 16.
    Gervasio, P.: Risoluzione delle equazioni alle derivate parziali con metodi spettrali in regioni partizionate in sottodomini. PhD thesis, Università di Milano, 1995Google Scholar
  17. 17.
    Goldfeld P. (2003): Balancing Neumann–Neumann preconditioners for the mixed formulation of almost-incompressible linear elasticity. Technical Report TR2003-847, Courant Institute of Mathematical Sciences, NYUGoogle Scholar
  18. 18.
    Goldfeld P., Pavarino L.F., Widlund O.B. (2003): Balancing Neumann–Neumann preconditioners for mixed approximations of heterogeneous problems in linear elasticity. Numer. Math. 95(2): 283–324MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Hughes, T.J.R.: Linear static and dynamic finite element analysis. (With the collaboration of Ferencz, R.M. and Raefsky, A.M.) In: The Finite Element Method. Prentice Hall, Englewood Cliffs (1987)Google Scholar
  20. 20.
    Klawonn A., Pavarino L.F. (1998): Overlapping Schwarz methods for mixed linear elasticity and Stokes problems. Comput. Methods Appl. Mech. Eng. 165, 233–245MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Klawonn, A., Rheinbach, O., Wohlmuth, B.: Dual-primal iterative substructuring for almost incompressible elasticity. Domain decomposition methods. In: Proceedings of the 16th International Conference on Domain Decomposition Methods, New York. Springer LNCSE (2006, to appear)Google Scholar
  22. 22.
    Klawonn A., Widlund O.B. (2001): FETI and Neumann–Neumann iterative substructuring methods: connections and new results. Commun. Pure Appl. Math. 54(1): 57–90MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Klawonn A., Widlund O.B. (2004): Dual-primal FETI methods for linear elasticity. Technical Report TR2004-855, Courant Institute of Mathematical Sciences, NYUGoogle Scholar
  24. 24.
    Le Tallec P., Mandel J., Vidrascu M. (1998): A Neumann–Neumann domain decomposition algorithm for solving plate and shell problems. SIAM J. Numer. Anal. 35(2): 836–867MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Le Tallec P., Patra A. (1997): Non-overlapping domain decomposition methods for adaptive hp approximations of the Stokes problem with discontinuous pressure fields. Comput. Methods Appl. Mech. Eng. 145, 361–379MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Li J. (2002): Dual-primal feti methods for stationary Stokes and Navier–Stokes equations. Technical Report TR2002-830, Courant Institute of Mathematical Sciences, NYUGoogle Scholar
  27. 27.
    Li J., Widlund O.B. (2005): BDDC algorithms for incompressible Stokes equations. Technical Report TR2005-861, Courant Institute of Mathematical Sciences, NYUGoogle Scholar
  28. 28.
    Li J., Widlund O.B. (2006): FETI–DP, BDDC, and block Cholesky methods. Int. J. Numer. Methods Eng. 66(2): 250–271MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Maday Y., Meiron D., Patera A., Rønquist E. (1993): Analysis of iterative methods for the steady and unsteady Stokes problem: application to spectral element discretizations. SIAM J. Sci. Comput. 14(2): 310–337MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Mandel J. (1993): Balancing domain decomposition. Commun. Numer. Methods Eng. 9(3): 233–241MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Mandel J., Brezina M. (1996): Balancing domain decomposition for problems with large jumps in coefficients. Math. Comput. 65(216): 1387–1401MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Mandel J., Dohrmann C.R. (2003): Convergence of a balancing domain decomposition by constraints and energy minimization. Numer. Linear Algebra Appl. 10(7): 639–659MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Mandel J., Dohrmann C.R., Tezaur R. (2005): An algebraic theory for primal and dual substructuring methods by constraints. Appl. Numer. Math. 54(2): 167–193MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Marini D., Quarteroni A. (1989): A relaxation procedure for domain decomposition methods using finite elements. Numer. Math. 55(5): 575–598MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Pasciak, J.E.: Two domain decomposition techniques for Stokes problems. In: Chan, T., Glowinski, R., Périaux, J., Widlund, O. (eds.) Domain Decomposition Methods. Philadelphia, pp. 419–430 (1989)Google Scholar
  36. 36.
    Pavarino L.F. (1997): Neumann–Neumann algorithms for spectral elements in three dimensions. RAIRO M 2  AN 31(4): 471–493MathSciNetGoogle Scholar
  37. 37.
    Pavarino L.F., Widlund O.B. (2000): Iterative substructuring methods for spectral element discretizations of elliptic systems. II: Mixed methods for linear elasticity and Stokes flow. SIAM J. Numer. Anal. 37(2): 375–402MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Pavarino L.F., Widlund O.B. (2002): Balancing Neumann–Neumann methods for incompressible Stokes equations. Commun. Pure Appl. Math. 55(3): 302–335MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Quarteroni, A.: Domain decomposition algorithms for the Stokes equations. In: Chan, T., Glowinski, R., Périaux, J., Widlund, O. (eds.) Domain Decomposition Methods. Philadelphia, pp. 431–442 (1989)Google Scholar
  40. 40.
    Quarteroni A., Valli A. (1999): Domain Decomposition Methods for Partial Differential Equations. Oxford Science Publications, New YorkMATHGoogle Scholar
  41. 41.
    Rønquist, E.: A domain decomposition solver for the steady Navier–Stokes equations. In: Ilin, A.V., Scott, L.R. (eds.) Proceedings of ICOSAHOM.95. Houston Journal of Mathematics Publication, pp. 469–485 (1996)Google Scholar
  42. 42.
    Rønquist, E.: Domain decomposition methods for the steady Stokes equations. In: Lai, C.-H., Bjørstad, P., Cross, M., Widlund, O. (eds.) Proceedings of DD11, pp. 326–336 (1999) (DDM.org)Google Scholar
  43. 43.
    Smith B.F., Bjørstad P., Gropp W.D. (1996): Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, cambridgeMATHGoogle Scholar
  44. 44.
    Stenberg R., Suri M. (1996): Mixed hp finite element methods for problems in elasticity and Stokes flow. Numer. Math. 72(3): 367–390MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Toselli A., Widlund O. (2005): Domain decomposition methods – algorithms and theory. Springer Series in Computational Mathematics, vol. 34. Springer, Berlin Heidelberg New YorkMATHGoogle Scholar
  46. 46.
    Toselli A.E. (2000): Neumann–Neumann methods for vector field problems. ETNA 11, 1–24MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • L. Beirão da Veiga
    • 1
    • 2
  • C. Lovadina
    • 2
    • 3
  • L. F. Pavarino
    • 1
  1. 1.Department of MathematicsUniversità di MilanoMilanoItaly
  2. 2.I.M.A.T.I.–C.N.R.PaviaItaly
  3. 3.Dipartimento di MatematicaUniversità di PaviaPaviaItaly

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