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A quasi-dual Lagrange multiplier space for serendipity mortar finite elements in 3D

Published online by Cambridge University Press:  15 February 2004

Bishnu P. Lamichhane  and
Barbara I. Wohlmuth
Bishnu P. Lamichhane
Affiliation:
Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Germany. [email protected].;[email protected].
Barbara I. Wohlmuth
Affiliation:
Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Germany. [email protected].;[email protected].
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Abstract

Domain decomposition techniques provide a flexible tool for the numerical approximation of partial differential equations. Here, we consider mortar techniques for quadratic finite elements in 3D with different Lagrange multiplier spaces. In particular, we focus on Lagrange multiplier spaces which yield optimal discretization schemes and a locally supported basis for the associated constrained mortar spaces in case of hexahedral triangulations. As a result, standard efficient iterative solvers as multigrid methods can be easily adapted to the nonconforming situation. We present the discretization errors in different norms for linear and quadratic mortar finite elements with different Lagrange multiplier spaces. Numerical results illustrate the performance of our approach.

Keywords

Mortar finite elementsLagrange multiplierdual spacedomain decompositionnonmatching triangulation.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 1 , January 2004 , pp. 73 - 92
Copyright
© EDP Sciences, SMAI, 2004

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References

Bastian, P., Birken, K., Johannsen, K., Lang, S., Neuß, N., Rentz–Reichert, H. and Wieners, C., UG – a flexible software toolbox for solving partial differential equations. Comput. Vis. Sci. 1 (1997) 27–40. CrossRef
Ben Belgacem, F., The mortar finite element method with Lagrange multipliers. Numer. Math. 84 (1999) 173–197. CrossRef
Ben Belgacem, F. and Maday, Y., The mortar element method for three dimensional finite elements. RAIRO Modél. Math. Anal. Numér. 31 (1997) 289–302. CrossRef
Bernardi, C., Debit, N. and Maday, Y., Coupling finite element and spectral methods: First results. Math. Comp. 54 (1990) 21–39. CrossRef
C. Bernardi, Y. Maday and A.T. Patera, Domain decomposition by the mortar element method, in Asymptotic and numerical methods for partial differential equations with critical parameters, H.G. Kaper and M. Garbey Eds., NATO ASI Series 39 (1993) 269–286.
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, H. Brezzi and J.-L. Lions Eds., Pitman, Paris (1994) 13–51.
Braess, D. and Dahmen, W., Stability estimates of the mortar finite element method for 3–dimensional problems. East–West J. Numer. Math. 6 (1998) 249–264.
Braess, D., Dahmen, W. and Wieners, C., A multigrid algorithm for the mortar finite element method. SIAM J. Numer. Anal. 37 (1999) 48–69. CrossRef
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York (1994).
F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag, New York (1991).
Brezzi, F. and Marini, D., Error estimates for the three-field formulation with bubble stabilization. Math. Comp 70 (2001) 911–934. CrossRef
F. Brezzi, L. Franca, D. Marini and A. Russo, Stabilization techniques for domain decomposition methods with non-matching grids, in Proc. of the 9th International Conference on Domain Decomposition, P. Bjørstad, M. Espedal and D. Keyes Eds., Domain Decomposition Press, Bergen (1998) 1–11.
Buffa, A., Error estimate for a stabilised domain decomposition method with nonmatching grids. Numer. Math. 90 (2002) 617–640. CrossRef
J. Gopalakrishnan, On the mortar finite element method. Ph.D. thesis, Texas A&M University (1999).
Kim, C., Lazarov, R.D., Pasciak, J.E. and Vassilevski, P.S., Multiplier spaces for the mortar finite element method in three dimensions. SIAM J. Numer. Anal. 39 (2000) 519–538. CrossRef
Lamichhane, B.P. and Wohlmuth, B.I., Higher order dual Lagrange multiplier spaces for mortar finite element discretizations. CALCOLO 39 (2002) 219–237. CrossRef
Seshaiyer, P. and Suri, M., Uniform hp convergence results for the mortar finite element method. Math. of Comput. 69 (2000) 521–546. CrossRef
Stevenson, R., Locally supported, piecewise polynomial biorthogonal wavelets on non-uniform meshes. Constr. Approx. 19 (2003) 477–508. CrossRef
C. Wieners and B.I. Wohlmuth, The coupling of mixed and conforming finite element discretizations, in Proc. of the 10th International Conference on Domain Decomposition, J. Mandel, C. Farhat and X. Cai Eds., AMS, Contemp. Math. (1998) 546–553.
C. Wieners and B.I. Wohlmuth, Duality estimates and multigrid analysis for saddle point problems arising from mortar discretizations. SISC 24 (2003) 2163–2184.
B.I. Wohlmuth, Discretization Methods and Iterative Solvers Based on Domain Decomposition. Lect. Notes Comput. Sci. 17, Springer, Heidelberg (2001).
Wohlmuth, B.I. and Krause, R.H., Multigrid methods based on the unconstrained product space arising from mortar finite element discretizations. SIAM J. Numer. Anal. 39 (2001) 192–213. CrossRef