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A residual based A POSTERIORI error estimator for an augmented mixed finite element methodin linear elasticity

Published online by Cambridge University Press:  16 January 2007

Tomás P. Barrios ,
Gabriel N. Gatica ,
María González  and
Norbert Heuer
Tomás P. Barrios
Affiliation:
Facultad de Ingeniería, Universidad Católica de la Santísima Concepción, Casilla 297, Concepción, Chile.
Gabriel N. Gatica
Affiliation:
GIMA, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile.
María González
Affiliation:
Departamento de Matemáticas, Universidade da Coruña, Campus de Elviña s/n, 15071 A Coruña, Spain.
Norbert Heuer
Affiliation:
BICOM and Department of Mathematical Sciences, Brunel University, Uxbridge, UB8 3PH, UK.
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Abstract

In this paper we develop a residual based a posteriori error analysis for an augmentedmixed 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 thecase of pure Dirichlet boundary conditions. In addition, several numericalexperiments confirming the theoretical properties of the estimator, andillustrating the capability of the corresponding adaptive algorithm to localize the singularities and the large stress regions of the solution, are also reported.

Keywords

Mixed finite elementaugmented formulationa posteriori error estimatorlinear elasticity.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 5 , September 2006 , pp. 843 - 869
Copyright
© EDP Sciences, SMAI, 2007

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References

Arnold, D.N., Brezzi, F. and Douglas, J., PEERS: A new mixed finite element method for plane elasticity. Japan J. Appl. Math. 1 (1984) 347–367. CrossRef
Braess, D., Klaas, O., Niekamp, R., Stein, E. and Wobschal, F., Error indicators for mixed finite elements in 2-dimensional linear elasticity. Comput. Method. Appl. M. 127 (1995) 345–356. CrossRef
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag (1991).
Carstensen, C., A posteriori error estimate for the mixed finite element method. Math. Comput. 66 (1997) 465–476. CrossRef
Carstensen, C. and Dolzmann, G., A posteriori error estimates for mixed FEM in elasticity. Numer. Math. 81 (1998) 187–209. CrossRef
P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, New York, Oxford (1978).
Clément, P., Approximation by finite element functions using local regularisation. RAIRO Anal. Numér. 9 (1975) 77–84.
Douglas, J. and Wan, J., An absolutely stabilized finite element method for the Stokes problem. Math. Comput. 52 (1989) 495–508. CrossRef
Gatica, G.N., 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.
Gatica, G.N., Analysis of a new augmented mixed finite element method for linear elasticity allowing $\mathbb{RT}_0-\mathbb{P}_1-\mathbb{P}_0$ approximations. ESAIM: M2AN 40 (2006) 1–28. CrossRef
Masud, A. and Hughes, T.J.R., A stabilized mixed finite element method for Darcy flow. Comput. Method. Appl. M. 191 (2002) 4341–4370. CrossRef
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).
Verfürth, R., A posteriori error estimation and adaptive mesh-refinement techniques. J. Comput. Appl. Math. 50 (1994) 67–83. CrossRef
R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner (Chichester) (1996).