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Nonconforming Discretization Techniques for Coupled Problems

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Multifield Problems in Solid and Fluid Mechanics

Part of the book series: Lecture Notes in Applied and Computational Mechanics ((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.

Research Project C12 “Nonconforming Discretization Techniques for Coupled Problems”

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References

  1. D. N. Arnold, D. Boffi, and R. S. Falk. Approximation by quadrilateral finite elements. Math. Comp., 71(239):909–922, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  3. F. Ben Belgacem. The mortar finite element method with Lagrange multipliers. Numer. Math., 84(2):173–197, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  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. F. Brezzi and M. Fortin. Mixed and hybrid finite element methods. Springer-Verlag, New York, 1991.

    MATH  Google Scholar 

  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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    MATH  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    MATH  MathSciNet  Google Scholar 

  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. B. Flemisch, J. M. Melenk, and B. I. Wohlmuth. Mortar methods with curved interfaces. Appl. Numer. Math., 54(3–4):339–361, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  21. J. Gopalakrishnan. On the mortar finite element method. PhD thesis, Texas A&M University, 1999.

    Google Scholar 

  22. P. Hauret. Numerical methods for the dynamic analysis of twoscale incompressible nonlinear structures. PhD thesis, Ecole Polytechnique, Paris, 2004.

    Google Scholar 

  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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  25. T. Hughes. The Finite Element Method. Prentice-Hall, New Jersey, 1987.

    MATH  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  31. B. P. Lamichhane and B. I. Wohlmuth. Mortar finite elements for interface problems. Computing, 72(3—4):333–348, 2004.

    MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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.

    Chapter  Google Scholar 

  35. T. A. Laursen. Computational contact and impact mechanics. Springer-Verlag, Berlin, 2002.

    MATH  Google Scholar 

  36. M. J. Lighthill. On sound generated aerodynamically. I. General theory. Proc. Roy. Soc. London. Ser. A., 211:564–587, 1952.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    MATH  MathSciNet  Google Scholar 

  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. 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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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. M. Puso. A 3D mortar method for solid mechanics. Internat. J. Numer. Methods Engrg., 59(3):315–336, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  48. B. I. Wohlmuth. A V-cycle multigrid approach for mortar finite elements. SIAM J. Numer. Anal., 42(6):2476–2495, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  49. P. Wriggers. Computational contact mechanics. Wiley, 2002.

    Google Scholar 

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Authors and Affiliations

  1. Institute for Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, 70569, Stuttgart, Germany

    Bernd Flemisch & Barbara I. Wohlmuth

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  1. Bernd Flemisch
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  2. Barbara I. Wohlmuth
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Editor information

Editors and Affiliations

  1. Institut für Wasserbau, Hydromechanik und Hydrosystemmodellierung, Universität Stuttgart, Pfaffenwaldring 61, 70569, Stuttgart, Germany

    Rainer Helmig

  2. Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117, Berlin, Germany

    Alexander Mielke

  3. Institut für Angewandte Analysis und Numerische Simulation, Universität Stuttgart, Pfaffenwaldring 57, 70569, Stuttgart, Germany

    Barbara I. Wohlmuth

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Flemisch, B., Wohlmuth, B.I. (2006). Nonconforming Discretization Techniques for Coupled Problems. In: Helmig, R., Mielke, A., Wohlmuth, B.I. (eds) Multifield Problems in Solid and Fluid Mechanics. Lecture Notes in Applied and Computational Mechanics, vol 28. Springer, Berlin, Heidelberg . https://doi.org/10.1007/978-3-540-34961-7_16

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  • DOI: https://doi.org/10.1007/978-3-540-34961-7_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-34959-4

  • Online ISBN: 978-3-540-34961-7

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