# A posteriori Error Estimates For the 3D Stabilized Mortar Finite Element Method applied to the Laplace Equation

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 37, Issue: 6, page 991-1011
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topBelhachmi, Zakaria. "A posteriori Error Estimates For the 3D Stabilized Mortar Finite Element Method applied to the Laplace Equation." ESAIM: Mathematical Modelling and Numerical Analysis 37.6 (2010): 991-1011. <http://eudml.org/doc/194201>.

@article{Belhachmi2010,

abstract = {
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.
},

author = {Belhachmi, Zakaria},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Mortar finite element method; a posteriori estimates;
mixed variational formulation; stabilization technique; non-matching grids.; error estimate; mortar finite element method; Laplace equation; decomposition techniques; mixed variational formulation; non-matching grids; stability},

language = {eng},

month = {3},

number = {6},

pages = {991-1011},

publisher = {EDP Sciences},

title = {A posteriori Error Estimates For the 3D Stabilized Mortar Finite Element Method applied to the Laplace Equation},

url = {http://eudml.org/doc/194201},

volume = {37},

year = {2010},

}

TY - JOUR

AU - Belhachmi, Zakaria

TI - A posteriori Error Estimates For the 3D Stabilized Mortar Finite Element Method applied to the Laplace Equation

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 37

IS - 6

SP - 991

EP - 1011

AB -
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.

LA - eng

KW - Mortar finite element method; a posteriori estimates;
mixed variational formulation; stabilization technique; non-matching grids.; error estimate; mortar finite element method; Laplace equation; decomposition techniques; mixed variational formulation; non-matching grids; stability

UR - http://eudml.org/doc/194201

ER -

## References

top- 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).
- 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.
- F. Ben Belgacem and Y. Maday, The mortar element method for three dimensional elements. RAIRO Modél. Anal. Numér.31 (1997) 289–302.
- C. Bernardi and F. Hecht, Error indicators for the mortar finite element discretization of the Laplace equation. Math. Comp.71 (2002) 1339–1370.
- C. Bernardi and V. Girault, A local regularization operator for triangular and quadrilateral finite elements. SIAM. J. Numer. Anal.35 (1998) 1893–1916
- C. Bernardi and Y. Maday, Mesh adaptivity in finite elements by the mortar method. Rev. Européeenne Élém. Finis9 (2000) 451–465.
- 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).
- 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.
- 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.
- V. Girault and P.A. Raviart, Finite Element Methods for the Navier–Stokes Equations. Springer-Verlag (1986).
- P.A. Raviart and J.M. Thomas, Primal hybrid finite element method for 2nd order elliptic equations. Math. Comp.31 (1977) 391–396.
- L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp.54 (1990) 483–493.
- R. Verfürth, Error estimates for some quasi-interpolation operators. Modél. Math. Anal. Numér.33 (1999) 695–713.
- R. Verfürth, A Review of A posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley & Teubner (1996).
- 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).
- B. Wohlmuth, A residual based error estimator for mortar finite element discretization. Numer. Math.84 (1999) 143–171.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.