A residual based a posteriori error estimator for an augmented mixed finite element method in linear elasticity
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 40 (2006) no. 5, p. 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 : https://doi.org/10.1051/m2an:2006036
Classification:  65N15,  65N30,  65N50,  74B05
@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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {40},
number = {5},
year = {2006},
pages = {843-869},
doi = {10.1051/m2an:2006036},
zbl = {1109.74047},
mrnumber = {2293249},
language = {en},
url = {http://www.numdam.org/item/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: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 40 (2006) no. 5, pp. 843-869. doi : 10.1051/m2an:2006036. http://www.numdam.org/item/M2AN_2006__40_5_843_0/

[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 0633.73074

[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 0860.73064

[3] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag (1991). | MR 1115205 | Zbl 0788.73002

[4] C. Carstensen, A posteriori error estimate for the mixed finite element method. Math. Comput. 66 (1997) 465-476. | Zbl 0864.65068

[5] C. Carstensen and G. Dolzmann, A posteriori error estimates for mixed FEM in elasticity. Numer. Math. 81 (1998) 187-209. | Zbl 0928.74093

[6] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, New York, Oxford (1978). | MR 520174 | Zbl 0383.65058

[7] P. Clément, Approximation by finite element functions using local regularisation. RAIRO Anal. Numér. 9 (1975) 77-84. | Numdam | Zbl 0368.65008

[8] J. Douglas and J. Wan, An absolutely stabilized finite element method for the Stokes problem. Math. Comput. 52 (1989) 495-508. | Zbl 0669.76051

[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 1065.65125

[10] G.N. Gatica, Analysis of a new augmented mixed finite element method for linear elasticity allowing ${\mathrm{ℝ𝕋}}_{0}-{ℙ}_{1}-{ℙ}_{0}$ approximations. ESAIM: M2AN 40 (2006) 1-28. | Numdam | Zbl pre05038390

[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 1015.76047

[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 1115239 | Zbl 0875.65090

[13] R. Verfürth, A posteriori error estimation and adaptive mesh-refinement techniques. J. Comput. Appl. Math. 50 (1994) 67-83. | Zbl 0811.65089

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