A quasi-dual Lagrange multiplier space for serendipity mortar finite elements in 3D
ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 1, pp. 73-92.

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.

DOI : 10.1051/m2an:2004004
Classification : 35N55, 65N30
Mots clés : Mortar finite elements, Lagrange multiplier, dual space, domain decomposition, nonmatching triangulation
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     title = {A quasi-dual {Lagrange} multiplier space for serendipity mortar finite elements in {3D}},
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Lamichhane, Bishnu P.; Wohlmuth, Barbara I. A quasi-dual Lagrange multiplier space for serendipity mortar finite elements in 3D. ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 1, pp. 73-92. doi : 10.1051/m2an:2004004. http://www.numdam.org/articles/10.1051/m2an:2004004/

[1] P. Bastian, K. Birken, K. Johannsen, S. Lang, N. Neuß, H. Rentz-Reichert and C. Wieners, UG - a flexible software toolbox for solving partial differential equations. Comput. Vis. Sci. 1 (1997) 27-40. | Zbl

[2] F. Ben Belgacem, The mortar finite element method with Lagrange multipliers. Numer. Math. 84 (1999) 173-197. | Zbl

[3] F. Ben Belgacem and Y. Maday, The mortar element method for three dimensional finite elements. RAIRO Modél. Math. Anal. Numér. 31 (1997) 289-302. | Numdam | Zbl

[4] C. Bernardi, N. Debit and Y. Maday, Coupling finite element and spectral methods: First results. Math. Comp. 54 (1990) 21-39. | Zbl

[5] 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. | Zbl

[6] 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. | Zbl

[7] D. Braess and W. Dahmen, Stability estimates of the mortar finite element method for 3-dimensional problems. East-West J. Numer. Math. 6 (1998) 249-264. | Zbl

[8] D. Braess, W. Dahmen and C. Wieners, A multigrid algorithm for the mortar finite element method. SIAM J. Numer. Anal. 37 (1999) 48-69. | Zbl

[9] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York (1994). | MR | Zbl

[10] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag, New York (1991). | MR | Zbl

[11] F. Brezzi and D. Marini, Error estimates for the three-field formulation with bubble stabilization. Math. Comp 70 (2001) 911-934. | Zbl

[12] 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.

[13] A. Buffa, Error estimate for a stabilised domain decomposition method with nonmatching grids. Numer. Math. 90 (2002) 617-640. | Zbl

[14] J. Gopalakrishnan, On the mortar finite element method. Ph.D. thesis, Texas A&M University (1999).

[15] 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 (2000) 519-538. | Zbl

[16] B.P. Lamichhane and B.I. Wohlmuth, Higher order dual Lagrange multiplier spaces for mortar finite element discretizations. CALCOLO 39 (2002) 219-237. | Zbl

[17] P. Seshaiyer and M. Suri, Uniform hp convergence results for the mortar finite element method. Math. of Comput. 69 (2000) 521-546. | Zbl

[18] R. Stevenson, Locally supported, piecewise polynomial biorthogonal wavelets on non-uniform meshes. Constr. Approx. 19 (2003) 477-508. | Zbl

[19] 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. | Zbl

[20] C. Wieners and B.I. Wohlmuth, Duality estimates and multigrid analysis for saddle point problems arising from mortar discretizations. SISC 24 (2003) 2163-2184. | Zbl

[21] B.I. Wohlmuth, Discretization Methods and Iterative Solvers Based on Domain Decomposition. Lect. Notes Comput. Sci. 17, Springer, Heidelberg (2001). | MR | Zbl

[22] B.I. Wohlmuth and R.H. Krause, Multigrid methods based on the unconstrained product space arising from mortar finite element discretizations. SIAM J. Numer. Anal. 39 (2001) 192-213. | Zbl

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