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.
Mots clés : Mortar finite elements, Lagrange multiplier, dual space, domain decomposition, nonmatching triangulation
@article{M2AN_2004__38_1_73_0,
author = {Lamichhane, Bishnu P. and Wohlmuth, Barbara I.},
title = {A quasi-dual {Lagrange} multiplier space for serendipity mortar finite elements in {3D}},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {73--92},
publisher = {EDP-Sciences},
volume = {38},
number = {1},
year = {2004},
doi = {10.1051/m2an:2004004},
mrnumber = {2073931},
zbl = {1105.65352},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2004004/}
}
TY - JOUR AU - Lamichhane, Bishnu P. AU - Wohlmuth, Barbara I. TI - A quasi-dual Lagrange multiplier space for serendipity mortar finite elements in 3D JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2004 SP - 73 EP - 92 VL - 38 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2004004/ DO - 10.1051/m2an:2004004 LA - en ID - M2AN_2004__38_1_73_0 ER -
%0 Journal Article %A Lamichhane, Bishnu P. %A Wohlmuth, Barbara I. %T A quasi-dual Lagrange multiplier space for serendipity mortar finite elements in 3D %J ESAIM: Modélisation mathématique et analyse numérique %D 2004 %P 73-92 %V 38 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2004004/ %R 10.1051/m2an:2004004 %G en %F M2AN_2004__38_1_73_0
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] ,,,,, and, UG - a flexible software toolbox for solving partial differential equations. Comput. Vis. Sci. 1 (1997) 27-40. | Zbl
[2] , The mortar finite element method with Lagrange multipliers. Numer. Math. 84 (1999) 173-197. | Zbl
[3] and, The mortar element method for three dimensional finite elements. RAIRO Modél. Math. Anal. Numér. 31 (1997) 289-302. | Numdam | Zbl
[4] , and, Coupling finite element and spectral methods: First results. Math. Comp. 54 (1990) 21-39. | Zbl
[5] , and, 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] , and, 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] and, Stability estimates of the mortar finite element method for 3-dimensional problems. East-West J. Numer. Math. 6 (1998) 249-264. | Zbl
[8] , and, A multigrid algorithm for the mortar finite element method. SIAM J. Numer. Anal. 37 (1999) 48-69. | Zbl
[9] and, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York (1994). | MR | Zbl
[10] and, Mixed and hybrid finite element methods. Springer-Verlag, New York (1991). | MR | Zbl
[11] and, Error estimates for the three-field formulation with bubble stabilization. Math. Comp 70 (2001) 911-934. | Zbl
[12] ,, and, 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] , Error estimate for a stabilised domain decomposition method with nonmatching grids. Numer. Math. 90 (2002) 617-640. | Zbl
[14] , On the mortar finite element method. Ph.D. thesis, Texas A&M University (1999).
[15] ,, and, Multiplier spaces for the mortar finite element method in three dimensions. SIAM J. Numer. Anal. 39 (2000) 519-538. | Zbl
[16] and, Higher order dual Lagrange multiplier spaces for mortar finite element discretizations. CALCOLO 39 (2002) 219-237. | Zbl
[17] and, Uniform hp convergence results for the mortar finite element method. Math. of Comput. 69 (2000) 521-546. | Zbl
[18] , Locally supported, piecewise polynomial biorthogonal wavelets on non-uniform meshes. Constr. Approx. 19 (2003) 477-508. | Zbl
[19] and, 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] and, Duality estimates and multigrid analysis for saddle point problems arising from mortar discretizations. SISC 24 (2003) 2163-2184. | Zbl
[21] , Discretization Methods and Iterative Solvers Based on Domain Decomposition. Lect. Notes Comput. Sci. 17, Springer, Heidelberg (2001). | MR | Zbl
[22] and, 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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