Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T19:56:06.867Z Has data issue: false hasContentIssue false
  • English
  • Français

A posteriori ErrorEstimates For the 3D Stabilized Mortar Finite Element Methodapplied to the Laplace Equation

Published online by Cambridge University Press:  15 November 2003

Zakaria Belhachmi
Zakaria Belhachmi*
Affiliation:
Laboratoire de Mathématiques LMAM, UMR7122, Université de Metz, Ile du Saulcy, 57045 Metz, France. belhach@poncelet.sciences.univ-metz.fr.
Get access

Abstract

We consider a non-conforming stabilized domaindecomposition technique forthe discretization of the three-dimensional Laplace equation.The aim is to extend the numerical analysis of residual error indicators tothis model problem. Two formulations of the problem are consideredand the error estimators are studied for both. In thefirst one, the error estimator provides upper and lower bounds forthe energy norm of the mortar finite element solution whereas inthe second case, it also estimates the error for the Lagrangemultiplier.

Keywords

Mortar finite element methoda posteriori estimatesmixed variational formulationstabilization techniquenon-matching grids.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 37 , Issue 6 , November 2003 , pp. 991 - 1011
Copyright
© EDP Sciences, SMAI, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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).
Ben Belgacem, F. and Brenner, S.C., Some nonstandard finite element estimates with applications to 3D Poisson and Signorini problems. Electron. Trans. Numer. Anal. 37 (2000) 1198–1216. CrossRef
Ben Belgacem, F. and Maday, Y., The mortar element method for three dimensional elements. RAIRO Modél. Anal. Numér. 31 (1997) 289–302. CrossRef
Bernardi, C. and Hecht, F., Error indicators for the mortar finite element discretization of the Laplace equation. Math. Comp. 71 (2002) 1339–1370.
Bernardi, C. and Girault, V., A local regularization operator for triangular and quadrilateral finite elements. SIAM. J. Numer. Anal. 35 (1998) 1893–1916 CrossRef
Bernardi, C. and Maday, Y., Mesh adaptivity in finite elements by the mortar method. Rev. Européeenne Élém. Finis 9 (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).
Raviart, P.A. and Thomas, J.M., Primal hybrid finite element method for 2nd order elliptic equations. Math. Comp. 31 (1977) 391–396.
Scott, L.R. and Zhang, S., Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483–493. CrossRef
Verfürth, R., Error estimates for some quasi-interpolation operators. Modél. Math. Anal. Numér. 33 (1999) 695–713. CrossRef
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).
Wohlmuth, B., A residual based error estimator for mortar finite element discretization. Numer. Math. 84 (1999) 143–171. CrossRef