no. 5
A residual based a posteriori error estimator for an augmented mixed finite element method in linear elasticity
ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 5, pp. 843-869.

In this paper we develop a residual based a posteriori error analysis for an augmented mixed finite element method applied to the problem of linear elasticity in the plane. More precisely, we derive a reliable and efficient a posteriori error estimator for the case of pure Dirichlet boundary conditions. In addition, several numerical experiments confirming the theoretical properties of the estimator, and illustrating the capability of the corresponding adaptive algorithm to localize the singularities and the large stress regions of the solution, are also reported.

DOI : 10.1051/m2an:2006036
Classification : 65N15, 65N30, 65N50, 74B05
Mots clés : mixed finite element, augmented formulation, a posteriori error estimator, linear elasticity 
@article{M2AN_2006__40_5_843_0,
     author = {Barrios, Tom\'as P. and Gatica, Gabriel N. and Gonz\'alez, Mar{\'\i}a and Heuer, Norbert},
     title = {A residual based a posteriori error estimator for an augmented mixed finite element method in linear elasticity},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {843--869},
     publisher = {EDP-Sciences},
     volume = {40},
     number = {5},
     year = {2006},
     doi = {10.1051/m2an:2006036},
     mrnumber = {2293249},
     zbl = {1109.74047},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2006036/}
}
TY  - JOUR
AU  - Barrios, Tomás P.
AU  - Gatica, Gabriel N.
AU  - González, María
AU  - Heuer, Norbert
TI  - A residual based a posteriori error estimator for an augmented mixed finite element method in linear elasticity
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2006
SP  - 843
EP  - 869
VL  - 40
IS  - 5
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2006036/
DO  - 10.1051/m2an:2006036
LA  - en
ID  - M2AN_2006__40_5_843_0
ER  - 
%0 Journal Article
%A Barrios, Tomás P.
%A Gatica, Gabriel N.
%A González, María
%A Heuer, Norbert
%T A residual based a posteriori error estimator for an augmented mixed finite element method in linear elasticity
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2006
%P 843-869
%V 40
%N 5
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2006036/
%R 10.1051/m2an:2006036
%G en
%F M2AN_2006__40_5_843_0
Barrios, Tomás P.; Gatica, Gabriel N.; González, María; Heuer, Norbert. A residual based a posteriori error estimator for an augmented mixed finite element method in linear elasticity. ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 5, pp. 843-869. doi : 10.1051/m2an:2006036. http://www.numdam.org/articles/10.1051/m2an:2006036/

[1] D.N. Arnold, F. Brezzi and J. Douglas, PEERS: A new mixed finite element method for plane elasticity. Japan J. Appl. Math. 1 (1984) 347-367. | Zbl

[2] D. Braess, O. Klaas, R. Niekamp, E. Stein and F. Wobschal, Error indicators for mixed finite elements in 2-dimensional linear elasticity. Comput. Method. Appl. M. 127 (1995) 345-356. | Zbl

[3] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag (1991). | MR | Zbl

[4] C. Carstensen, A posteriori error estimate for the mixed finite element method. Math. Comput. 66 (1997) 465-476. | Zbl

[5] C. Carstensen and G. Dolzmann, A posteriori error estimates for mixed FEM in elasticity. Numer. Math. 81 (1998) 187-209. | Zbl

[6] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, New York, Oxford (1978). | MR | Zbl

[7] P. Clément, Approximation by finite element functions using local regularisation. RAIRO Anal. Numér. 9 (1975) 77-84. | Numdam | Zbl

[8] J. Douglas and J. Wan, An absolutely stabilized finite element method for the Stokes problem. Math. Comput. 52 (1989) 495-508. | Zbl

[9] G.N. Gatica, A note on the efficiency of residual-based a posteriori error estimators for some mixed finite element methods. Electronic Trans. Numer. Anal. 17 (2004) 218-233. | Zbl

[10] G.N. Gatica, Analysis of a new augmented mixed finite element method for linear elasticity allowing ℝ𝕋 0 -ℙ 1 -ℙ 0 approximations. ESAIM: M2AN 40 (2006) 1-28. | Numdam

[11] A. Masud and T.J.R. Hughes, A stabilized mixed finite element method for Darcy flow. Comput. Method. Appl. M. 191 (2002) 4341-4370. | Zbl

[12] J.E. Roberts and J.-M. Thomas, Mixed and Hybrid Methods, in Handbook of Numerical Analysis II, Finite Element Methods (Part 1) P.G. Ciarlet and J.L. Lions Eds., North-Holland, Amsterdam (1991). | MR | Zbl

[13] R. Verfürth, A posteriori error estimation and adaptive mesh-refinement techniques. J. Comput. Appl. Math. 50 (1994) 67-83. | Zbl

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

Cité par Sources :