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”
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D. N. Arnold, D. Boffi, and R. S. Falk. Approximation by quadrilateral finite elements. Math. Comp., 71(239):909–922, 2002.
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.
F. Ben Belgacem. The mortar finite element method with Lagrange multipliers. Numer. Math., 84(2):173–197, 1999.
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.
F. Brezzi and M. Fortin. Mixed and hybrid finite element methods. Springer-Verlag, New York, 1991.
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.
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.
R. Dautray and J.-L. Lions. Mathematical analysis and numerical methods for science and technology. Vol. 5: Evolution problems I. Springer-Verlag, Berlin, 1992.
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.
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.
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.
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.
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.
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.
B. Flemisch, J. M. Melenk, and B. I. Wohlmuth. Mortar methods with curved interfaces. Appl. Numer. Math., 54(3–4):339–361, 2005.
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.
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.
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.
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.
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.
J. Gopalakrishnan. On the mortar finite element method. PhD thesis, Texas A&M University, 1999.
P. Hauret. Numerical methods for the dynamic analysis of twoscale incompressible nonlinear structures. PhD thesis, Ecole Polytechnique, Paris, 2004.
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.
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.
T. Hughes. The Finite Element Method. Prentice-Hall, New Jersey, 1987.
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.
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.
B. P. Lamichhane. Higher order mortar finite elements with dual Lagrange multiplier spaces and applications. PhD thesis, University of Stuttgart, 2006.
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.
B. P. Lamichhane and B. I. Wohlmuth. Higher order dual Lagrange multiplier spaces for mortar finite element discretizations. Calcolo, 39(4):219–237, 2002.
B. P. Lamichhane and B. I. Wohlmuth. Mortar finite elements for interface problems. Computing, 72(3—4):333–348, 2004.
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.
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.
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.
T. A. Laursen. Computational contact and impact mechanics. Springer-Verlag, Berlin, 2002.
M. J. Lighthill. On sound generated aerodynamically. I. General theory. Proc. Roy. Soc. London. Ser. A., 211:564–587, 1952.
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.
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.
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.
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.
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.
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.
M. Puso. A 3D mortar method for solid mechanics. Internat. J. Numer. Methods Engrg., 59(3):315–336, 2004.
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.
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.
B. I. Wohlmuth. A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J. Numer. Anal., 38(3):989–1012, 2000.
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.
B. I. Wohlmuth. A V-cycle multigrid approach for mortar finite elements. SIAM J. Numer. Anal., 42(6):2476–2495, 2005.
P. Wriggers. Computational contact mechanics. Wiley, 2002.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
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
Download citation
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
eBook Packages: EngineeringEngineering (R0)