Advertisement

Nonconforming Discretization Techniques for Coupled Problems

  • Bernd Flemisch
  • Barbara I. Wohlmuth
Part of the Lecture Notes in Applied and Computational Mechanics book series (LNACM, volume 28)

Summary

Multifield problems yield coupled problem formulations for which nonconforming discretizations schemes and problem-adapted solvers can be used to develop efficient numerical algorithms. Of crucial importance are numerically robust transmission operators based on weak continuity conditions. This paper presents the construction of such operators by means of dual discrete Lagrange multipliers for higher order discretizations and for general quadrilateral triangulations of possibly curved interfaces. Various applications are considered, including aero-acoustics, elasto-acoustics, contact and heat transfer.

Keywords

Domain decomposition non-matching grids dual Lagrange multipliers mortar finite elements nonconforming discretizations 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    D. N. Arnold, D. Boffi, and R. S. Falk. Approximation by quadrilateral finite elements. Math. Comp., 71(239):909–922, 2002.zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    A. Bamberger, R. Glowinski, and Q. H. Tran. A domain decomposition method for the acoustic wave equation with discontinuous coefficients and grid change. SIAM J. Numer. Anal., 34(2):603–639, 1997.zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    F. Ben Belgacem. The mortar finite element method with Lagrange multipliers. Numer. Math., 84(2):173–197, 1999.zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    C. Bernardi, Y. Maday, and A. T. Patera. A new nonconforming approach to domain decomposition: the mortar element method. In Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. XI (Paris, 1989–1991), volume 299 of Pitman Res. Notes Math. Ser., pages 13–51. Longman Sci. Tech., Harlow, 1994.Google Scholar
  5. 5.
    F. Brezzi and M. Fortin. Mixed and hybrid finite element methods. Springer-Verlag, New York, 1991.zbMATHGoogle Scholar
  6. 6.
    K. S. Chavan, B. P. Lamichhane, and B. I. Wohlmuth. Locking-free finite element methods for linear and nonlinear elasticity in 2D and 3D. Technical Report 13, University of Stuttgart, SFB 404, 2005. To appear in Comp. Meth. Appl. Mech. Engrg.Google Scholar
  7. 7.
    P. Ciarlet, Jr., J. Huang, and J. Zou. Some observations on generalized saddle-point problems. SIAM J. Matrix Anal. Appl., 25(1):224–236, 2003.zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    R. Dautray and J.-L. Lions. Mathematical analysis and numerical methods for science and technology. Vol. 5: Evolution problems I. Springer-Verlag, Berlin, 1992.zbMATHGoogle Scholar
  9. 9.
    M. Dryja, A. Gantner, O. B. Widlund, and B. I. Wohlmuth. Multilevel additive Schwarz preconditioner for nonconforming mortar finite element methods. J. Numer. Math., 12(1):23–38, 2004.zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    C. Eck and B. Wohlmuth. Convergence of a contact-Neumann iteration for the solution of two-body contact problems. Math. Models Methods Appl. Sci., 13(8):1103–1118, 2003.zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    B. Flemisch, M. Kaltenbacher, and B. I. Wohlmuth. Elasto-acoustic and acoustic-acoustic coupling on nonmatching grids. Technical Report 10, University of Stuttgart, SFB 404, 2005. To appear in Internat. J. Numer. Methods Engrg.Google Scholar
  12. 12.
    B. Flemisch, Y. Maday, F. Rapetti, and B. Wohlmuth. Coupling scalar and vector potentials on nonmatching grids for eddy currents in a moving conductor. J. Comput. Appl. Math., 168(1–2):191–205, 2004.zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    B. Flemisch, Y. Maday, F. Rapetti, and B. Wohlmuth. Scalar and vector potentials’ coupling on nonmatching grids for the simulation of an electromagnetic brake. COMPEL, 24(3):1061–1070, 2005.zbMATHMathSciNetGoogle Scholar
  14. 14.
    B. Flemisch, M. Mair, and B. I. Wohlmuth. Nonconforming discretization techniques for overlapping domain decompositions. In M. Feistauer et al., editors, Numerical mathematics and advanced applications. Proceedings of Enumath 2003, Prague, Czech Republic, August 18–22, 2003, pages 316–325. Springer, Berlin, 2004.Google Scholar
  15. 15.
    B. Flemisch, J. M. Melenk, and B. I. Wohlmuth. Mortar methods with curved interfaces. Appl. Numer. Math., 54(3–4):339–361, 2005.zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    B. Flemisch, M. A. Puso, and B. I. Wohlmuth. A new dual mortar method for curved interfaces: 2D elasticity. Internat. J. Numer. Methods Engrg., 63(6):813–832, 2005.zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    B. Flemisch and B. I. Wohlmuth. A domain decomposition method on nested domains and nonmatching grids. Numer. Methods Partial Differential Equations, 20(3):374–387, 2004.zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    B. Flemisch and B. I. Wohlmuth. Nonconforming methods for nonlinear elasticity problems. Technical Report 03, University of Stuttgart, SFB 404, 2005. To appear in the Proceedings of the 16th International Conference on Domain Decomposition Methods.Google Scholar
  19. 19.
    B. Flemisch and B. I. Wohlmuth. Stable Lagrange multipliers for quadrilateral meshes of curved interfaces in 3D, IANS preprint 2005/005. Technical report, University of Stuttgart, 2005. To appear in Comp. Meth. Appl. Mech. Engrg.Google Scholar
  20. 20.
    R. Glowinski, J. He, A. Lozinski, J. Rappaz, and J. Wagner. Finite element approximation of multi-scale elliptic problems using patches of elements. Numer. Math., 101(4):663–687, 2005.zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    J. Gopalakrishnan. On the mortar finite element method. PhD thesis, Texas A&M University, 1999.Google Scholar
  22. 22.
    P. Hauret. Numerical methods for the dynamic analysis of twoscale incompressible nonlinear structures. PhD thesis, Ecole Polytechnique, Paris, 2004.Google Scholar
  23. 23.
    P. Hauret and P. L. Tallec. Dirichlet-Neumann preconditioners for elliptic problems with small disjoint geometric refinements on the boundary. Technical Report 552, CMAP — Ecole Polytechnique, 2004.Google Scholar
  24. 24.
    S. Hüeber and B. I. Wohlmuth. A primal-dual active set strategy for non-linear multibody contact problems. Comput. Methods Appl. Mech. Engrg., 194:3147–3166, 2005.zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    T. Hughes. The Finite Element Method. Prentice-Hall, New Jersey, 1987.zbMATHGoogle Scholar
  26. 26.
    C. Kim, R. D. Lazarov, J. E. Pasciak, and P. S. Vassilevski. Multiplier spaces for the mortar finite element method in three dimensions. SIAM J. Numer. Anal., 39(2):519–538, 2001.zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    R. H. Krause and B. I. Wohlmuth. A Dirichlet-Neumann type algorithm for contact problems with friction. Comput. Vis. Sci., 5(3):139–148, 2002.zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    B. P. Lamichhane. Higher order mortar finite elements with dual Lagrange multiplier spaces and applications. PhD thesis, University of Stuttgart, 2006.Google Scholar
  29. 29.
    B. P. Lamichhane, R. P. Stevenson, and B. I. Wohlmuth. Higher order mortar finite element methods in 3D with dual Lagrange multiplier bases. Numer. Math., 102(1):93–121, 2005.zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    B. P. Lamichhane and B. I. Wohlmuth. Higher order dual Lagrange multiplier spaces for mortar finite element discretizations. Calcolo, 39(4):219–237, 2002.zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    B. P. Lamichhane and B. I. Wohlmuth. Mortar finite elements for interface problems. Computing, 72(3—4):333–348, 2004.zbMATHMathSciNetGoogle Scholar
  32. 32.
    B. P. Lamichhane and B. I. Wohlmuth. A quasi-dual Lagrange multiplier space for serendipity mortar finite elements in 3D. M2AN Math. Model. Numer. Anal., 38(1):73–92, 2004.zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    B. P. Lamichhane and B. I. Wohlmuth. Biorthogonal bases with local support and approximation properties. Technical Report 02, University of Stuttgart, SFB 404, 2005. To appear in Math. Comp.Google Scholar
  34. 34.
    B. P. Lamichhane and B. I. Wohlmuth. Mortar finite elements with dual Lagrange multipliers: some applications. In Kornhuber, Ralf (ed.) et al., Domain decomposition methods in science and engineering. Selected papers of the 15th International Conference on Domain Decomposition, Berlin, Germany, July 21–25, 2003, pages 319–326. Springer, Berlin, 2005.CrossRefGoogle Scholar
  35. 35.
    T. A. Laursen. Computational contact and impact mechanics. Springer-Verlag, Berlin, 2002.zbMATHGoogle Scholar
  36. 36.
    M. J. Lighthill. On sound generated aerodynamically. I. General theory. Proc. Roy. Soc. London. Ser. A., 211:564–587, 1952.zbMATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    Y. Maday, F. Rapetti, and B. I. Wohlmuth. Coupling between scalar and vector potentials by the mortar element method. C. R. Math. Acad. Sci. Paris, 334(10):933–938, 2002.zbMATHMathSciNetGoogle Scholar
  38. 38.
    Y. Maday, F. Rapetti, and B. I. Wohlmuth. The influence of quadrature formulas in 2D and 3D mortar element methods. In Recent developments in domain decomposition methods (Zürich, 2001), volume 23 of Lect. Notes Comput. Sci. Eng., pages 203–221. Springer, Berlin, 2002.Google Scholar
  39. 39.
    Y. Maday, F. Rapetti, and B. I. Wohlmuth. Mortar element coupling between global scalar and local vector potentials to solve eddy current problems. In F. Brezzi et al., editors, Numerical mathematics and advanced applications. Proceedings of Enumath 2001, Ischia, July 2001, pages 847–865. Springer, Berlin, 2003.Google Scholar
  40. 40.
    M. Mair and B. I. Wohlmuth. A domain decomposition method for domains with holes using a complementary decomposition. Comput. Methods Appl. Mech. Engrg., 193(45–47):4961–4978, 2004.zbMATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    T. W. McDevitt and T. A. Laursen. A mortar-finite element formulation for frictional contact problems. Internat. J. Numer. Methods Engrg., 48(10):1525–1547, 2000.zbMATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    P. Oswald and B. I. Wohlmuth. On polynomial reproduction of dual FE bases. In Domain decomposition methods in science and engineering (Lyon, 2000), Theory Eng. Appl. Comput. Methods, pages 85–96. Internat. Center Numer. Methods Eng. (CIMNE), Barcelona, 2002.Google Scholar
  43. 43.
    M. Puso. A 3D mortar method for solid mechanics. Internat. J. Numer. Methods Engrg., 59(3):315–336, 2004.zbMATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    E. Stein and M. Rüter. Finite element methods for elasticity with error-controlled discretization and model adaptivity. In E. Stein, R. de Borst and T.J.R. Hughes, editors, Encyclopedia of Computational Mechanics, pages 5–58. Wiley, Chichester, 2004.Google Scholar
  45. 45.
    C. Wieners and B. I. Wohlmuth. Duality estimates and multigrid analysis for saddle point problems arising from mortar discretizations. SIAM J. Sci. Comput., 24(6):2163–2184, 2003.zbMATHMathSciNetCrossRefGoogle Scholar
  46. 46.
    B. I. Wohlmuth. A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J. Numer. Anal., 38(3):989–1012, 2000.zbMATHMathSciNetCrossRefGoogle Scholar
  47. 47.
    B. I. Wohlmuth. A comparison of dual Lagrange multiplier spaces for mortar finite element discretizations. M2AN Math. Model. Numer. Anal., 36(6):995–1012, 2002.zbMATHMathSciNetCrossRefGoogle Scholar
  48. 48.
    B. I. Wohlmuth. A V-cycle multigrid approach for mortar finite elements. SIAM J. Numer. Anal., 42(6):2476–2495, 2005.zbMATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    P. Wriggers. Computational contact mechanics. Wiley, 2002.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Bernd Flemisch
    • 1
  • Barbara I. Wohlmuth
    • 1
  1. 1.Institute for Applied Analysis and Numerical SimulationUniversity of StuttgartStuttgartGermany

Personalised recommendations