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

Zakaria Belhachmi

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

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

Abstract

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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.

How to cite

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Belhachmi, 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

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  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).  
  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.  
  3. F. Ben Belgacem and Y. Maday, The mortar element method for three dimensional elements. RAIRO Modél. Anal. Numér.31 (1997) 289–302.  Zbl0868.65082
  4. C. Bernardi and F. Hecht, Error indicators for the mortar finite element discretization of the Laplace equation. Math. Comp.71 (2002) 1339–1370.  Zbl1012.65108
  5. C. Bernardi and V. Girault, A local regularization operator for triangular and quadrilateral finite elements. SIAM. J. Numer. Anal.35 (1998) 1893–1916 Zbl0913.65007
  6. C. Bernardi and Y. Maday, Mesh adaptivity in finite elements by the mortar method. Rev. Européeenne Élém. Finis9 (2000) 451–465.  Zbl0954.65081
  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).  Zbl0797.65094
  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.  Zbl0875.65086
  10. V. Girault and P.A. Raviart, Finite Element Methods for the Navier–Stokes Equations. Springer-Verlag (1986).  Zbl0585.65077
  11. P.A. Raviart and J.M. Thomas, Primal hybrid finite element method for 2nd order elliptic equations. Math. Comp.31 (1977) 391–396.  Zbl0364.65082
  12. L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp.54 (1990) 483–493.  Zbl0696.65007
  13. R. Verfürth, Error estimates for some quasi-interpolation operators. Modél. Math. Anal. Numér.33 (1999) 695–713.  Zbl0938.65125
  14. R. Verfürth, A Review of A posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley & Teubner (1996).  Zbl0853.65108
  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).  
  16. B. Wohlmuth, A residual based error estimator for mortar finite element discretization. Numer. Math.84 (1999) 143–171.  Zbl0962.65090

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