A posteriori error estimates for the 3D stabilized Mortar finite element method applied to the Laplace equation
ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 6, pp. 991-1011.

We consider a non-conforming stabilized domain decomposition technique for the discretization of the three-dimensional Laplace equation. The aim is to extend the numerical analysis of residual error indicators to this model problem. Two formulations of the problem are considered and the error estimators are studied for both. In the first one, the error estimator provides upper and lower bounds for the energy norm of the mortar finite element solution whereas in the second case, it also estimates the error for the Lagrange multiplier.

DOI : 10.1051/m2an:2003064
Classification : 65N30
Mots clés : Mortar finite element method, a posteriori estimates, mixed variational formulation, stabilization technique, non-matching grids
@article{M2AN_2003__37_6_991_0,
     author = {Belhachmi, Zakaria},
     title = {A posteriori error estimates for the $3${D} stabilized {Mortar} finite element method applied to the {Laplace} equation},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {991--1011},
     publisher = {EDP-Sciences},
     volume = {37},
     number = {6},
     year = {2003},
     doi = {10.1051/m2an:2003064},
     mrnumber = {2026405},
     zbl = {1076.65092},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2003064/}
}
TY  - JOUR
AU  - Belhachmi, Zakaria
TI  - A posteriori error estimates for the $3$D stabilized Mortar finite element method applied to the Laplace equation
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2003
SP  - 991
EP  - 1011
VL  - 37
IS  - 6
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2003064/
DO  - 10.1051/m2an:2003064
LA  - en
ID  - M2AN_2003__37_6_991_0
ER  - 
%0 Journal Article
%A Belhachmi, Zakaria
%T A posteriori error estimates for the $3$D stabilized Mortar finite element method applied to the Laplace equation
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2003
%P 991-1011
%V 37
%N 6
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2003064/
%R 10.1051/m2an:2003064
%G en
%F M2AN_2003__37_6_991_0
Belhachmi, Zakaria. A posteriori error estimates for the $3$D stabilized Mortar finite element method applied to the Laplace equation. ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 6, pp. 991-1011. doi : 10.1051/m2an:2003064. http://www.numdam.org/articles/10.1051/m2an:2003064/

[1] F. Ben Belgacem, A stabilized domain decomposition method with non-matching grids to the Stokes problem in three dimensions. SIAM. J. Numer. Anal. (to appear). | MR

[2] F. Ben Belgacem and S.C. Brenner, Some nonstandard finite element estimates with applications to 3D Poisson and Signorini problems. Electron. Trans. Numer. Anal. 37 (2000) 1198-1216. | Zbl

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

[4] C. Bernardi and F. Hecht, Error indicators for the mortar finite element discretization of the Laplace equation. Math. Comp. 71 (2002) 1339-1370. | Zbl

[5] C. Bernardi and V. Girault, A local regularization operator for triangular and quadrilateral finite elements. SIAM. J. Numer. Anal. 35 (1998) 1893-1916 | Zbl

[6] C. Bernardi and Y. Maday, Mesh adaptivity in finite elements by the mortar method. Rev. Européeenne Élém. Finis 9 (2000) 451-465. | Zbl

[7] C. Bernardi, Y. Maday and A.T. Patera, A New Non Conforming Approach to Domain Decomposition: The Mortar Element Method. Collège de France Seminar, Pitman, H. Brezis, J.-L. Lions (1990). | Zbl

[8] F. Brezzi, L.P. Franca, D. Marini and A. Russo, Stabilization techniques for domain decomposition with non-matching grids, Domain Decomposition Methods in Sciences and Engineering, P. Bjostrad, M. Espedal, D. Keyes Eds., Domain Decomposition Press, Bergen (1998) 1-11.

[9] P.G. Ciarlet, Basic error estimates for elliptic problems, in The Handbook of Numerical Analysis, Vol. II, P.G. Ciarlet, J.-L. Lions Eds., North-Holland (1991) 17-351. | Zbl

[10] V. Girault and P.A. Raviart, Finite Element Methods for the Navier-Stokes Equations. Springer-Verlag (1986). | Zbl

[11] P.A. Raviart and J.M. Thomas, Primal hybrid finite element method for 2nd order elliptic equations. Math. Comp. 31 (1977) 391-396. | Zbl

[12] L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483-493. | Zbl

[13] R. Verfürth, Error estimates for some quasi-interpolation operators. Modél. Math. Anal. Numér. 33 (1999) 695-713. | Numdam | Zbl

[14] R. Verfürth, A Review of A posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley & Teubner (1996). | Zbl

[15] O.B. Widlund, An extention theorem for finite element spaces with three applications, in Numerical Techniques in Continuum Mechanics, Proceedings of the Second GAMM Seminar, W Hackbush, K. Witsch Eds., Kiel (1986). | Zbl

[16] B. Wohlmuth, A residual based error estimator for mortar finite element discretization. Numer. Math. 84 (1999) 143-171. | Zbl

Cité par Sources :