A posteriori error estimates for the $3$D stabilized Mortar finite element method applied to the Laplace equation
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 6, p. 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 : https://doi.org/10.1051/m2an:2003064
Classification:  65N30
Keywords: 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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {37},
number = {6},
year = {2003},
pages = {991-1011},
doi = {10.1051/m2an:2003064},
zbl = {1076.65092},
mrnumber = {2026405},
language = {en},
url = {http://www.numdam.org/item/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: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 6, pp. 991-1011. doi : 10.1051/m2an:2003064. http://www.numdam.org/item/M2AN_2003__37_6_991_0/

[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 2084231 | Zbl pre02139895

[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 0974.74055

[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 0868.65082

[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 1012.65108

[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 0913.65007

[6] C. Bernardi and Y. Maday, Mesh adaptivity in finite elements by the mortar method. Rev. Européeenne Élém. Finis 9 (2000) 451-465. | Zbl 0954.65081

[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 0797.65094

[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 0875.65086

[10] V. Girault and P.A. Raviart, Finite Element Methods for the Navier-Stokes Equations. Springer-Verlag (1986). | Zbl 0585.65077

[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 0364.65082

[12] L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483-493. | Zbl 0696.65007

[13] R. Verfürth, Error estimates for some quasi-interpolation operators. Modél. Math. Anal. Numér. 33 (1999) 695-713. | Numdam | Zbl 0938.65125

[14] R. Verfürth, A Review of A posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley & Teubner (1996). | Zbl 0853.65108

[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 0615.65114

[16] B. Wohlmuth, A residual based error estimator for mortar finite element discretization. Numer. Math. 84 (1999) 143-171. | Zbl 0962.65090