We present a new stabilized mixed finite element method for the linear elasticity problem in . The approach is based on the introduction of Galerkin least-squares terms arising from the constitutive and equilibrium equations, and from the relation defining the rotation in terms of the displacement. We show that the resulting augmented variational formulation and the associated Galerkin scheme are well posed, and that the latter becomes locking-free and asymptotically locking-free for Dirichlet and mixed boundary conditions, respectively. In particular, the discrete scheme allows the utilization of Raviart-Thomas spaces of lowest order for the stress tensor, piecewise linear elements for the displacement, and piecewise constants for the rotation. In the case of mixed boundary conditions, the essential one (Neumann) is imposed weakly, which yields the introduction of the trace of the displacement as a suitable Lagrange multiplier. This trace is then approximated by piecewise linear elements on an independent partition of the Neumann boundary whose mesh size needs to satisfy a compatibility condition with the mesh size associated to the triangulation of the domain. Several numerical results illustrating the good performance of the augmented mixed finite element scheme in the case of Dirichlet boundary conditions are also reported.
@article{M2AN_2006__40_1_1_0,
author = {Gatica, Gabriel N.},
title = {Analysis of a new augmented mixed finite element method for linear elasticity allowing $\mathbb {RT}\_0 - \mathbb {P}\_1 - \mathbb {P}\_0$ approximations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {40},
year = {2006},
pages = {1-28},
doi = {10.1051/m2an:2006003},
mrnumber = {2223502},
zbl = {pre05038390},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2006__40_1_1_0}
}
Gatica, Gabriel 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: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 40 (2006) pp. 1-28. doi : 10.1051/m2an:2006003. http://gdmltest.u-ga.fr/item/M2AN_2006__40_1_1_0/
[1] , and, PEERS: A new mixed finite element method for plane elasticity. Japan J. Appl. Math. 1 (1984) 347-367. | Zbl 0633.73074
[2] , and, A stable finite element method for the Stokes equations. Calcolo 21 (1984) 337-344. | Zbl 0593.76039
[3] , and, A family of higher order mixed finite element methods for plane elasticity. Numer. Math. 45 (1984) 1-22. | Zbl 0558.73066
[4] and, Well-posedness of the fundamental boundary value problems for constrained anisotropic elastic materials. Arch. Rational Mech. Analysis 98 (1987) 143-190. | Zbl 0618.73012
[5] and, On the mixed finite element method with Lagrange multipliers. Numer. Methods Partial Differential Equations 19 (2003) 192-210. | Zbl 1021.65056
[6] , and, A mixed finite element method for nonlinear elasticity: two-fold saddle point approach and a posteriori error estimate. Numer. Math. 91 (2002) 197-222. | Zbl 1067.74062
[7] , and, A wavelet-based stabilization of the mixed finite element method with Lagrange multipliers. Appl. Math. Lett. (in press). | MR 2202412 | Zbl 1101.65106
[8] , Finite Elements. Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press (1997). | MR 1463151 | Zbl 0894.65054
[9] and, Stabilized mixed methods for the Stokes problem. Numer. Math. 53 (1988) 225-235. | Zbl 0669.76052
[10] and, Mixed and Hybrid Finite Element Methods. Springer-Verlag (1991). | MR 1115205 | Zbl 0788.73002
[11] and, A minimal stabilisation procedure for mixed finite element methods. Numer. Math. 89 (2001) 457-491. | Zbl 1009.65067
[12] , and, Variable degree mixed methods for second order elliptic problems. Mat. Apl. Comput. 4 (1985) 19-34. | Zbl 0592.65073
[13] , and, Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47 (1985) 217-235. | Zbl 0599.65072
[14] , and, Mixed finite element methods with continuous stresses. Math. Models Methods Appl. Sci. 3 (1993) 275-287. | Zbl 0774.73066
[15] and, Locking-free mixed stabilized finite element methods for bending-dominated shells, in Plates and shells (Quebec, QC, 1996), American Mathematical Society, Providence, RI, CRM Proceedings Lecture Notes 21 (1999) 81-94. | Zbl 0958.74060
[16] , The Finite Element Method for Elliptic Problems. North-Holland (1978). | MR 520174 | Zbl 0383.65058
[17] and, An absolutely stabilized finite element method for the Stokes problem. Math. Comput. 52 (1989) 495-508. | Zbl 0669.76051
[18] and, Analysis of some stabilized low-order mixed finite element methods for Reissner-Mindlin plates. Comput. Methods Appl. Mech. Engrg. 191 (2001) 157-179. | Zbl 1041.74067
[19] , New Mixed Finite Element Methods. Ph.D. Thesis, Stanford University (1987).
[20] and, Two classes of finite element methods. Comput. Methods Appl. Mech. Engrg. 69 (1988) 89-129. | Zbl 0629.73053
[21] and, Unlocking with residual-free bubbles. Comput. Methods Appl. Mech. Engrg. 142 (1997) 361-364. | Zbl 0890.73064
[22] and, Error analysis of Galerkin least squares methods for the elasticity equations. SIAM J. Numer. Anal. 28 (1991) 1680-1697. | Zbl 0759.73055
[23] , Korn's inequalities and their applications in continuum mechanics. SIAM Rev. 37 (1995) 491-511. | Zbl 0840.73010
[24] and, Eigenvalue problems associated with Korn's inequalities. Arch. Rational Mech. Anal. 40 (1971) 384-402. | Zbl 0223.73011
[25] and, On inequalities of Korn, Friedrichs and Babuška-Aziz. Arch. Rational Mech. Analysis 82 (1983) 165-179. | Zbl 0512.73017
[26] and, Analysis of locally stabilized mixed finite element methods for the Stokes problem. Math. Comput. 58 (1992) 1-10. | Zbl 0738.76040
[27] and, Stabilized mixed finite element approximations of incompressible flow problems. Zeitschrift für Angewandte Mathematik und Mechanik 72 (1992) T483-T486. | Zbl 0766.76050
[28] , Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press (2000). | MR 1742312 | Zbl 0948.35001
[29] and, A stabilized mixed finite element method for Darcy flow. Comput. Methods Appl. Mech. Engrg. 191 (2002) 4341-4370. | Zbl 1015.76047
[30] and, A new locking-free equilibrium mixed element for plane elasticity with continuous displacement interpolation. Comput. Methods Appl. Mech. Engrg 191 (2002) 1843-1860. | Zbl 1098.74708
[31] and, Fourier analysis of stabilized - mixed finite element approximation. SIAM J. Numer. Anal. 39 (2001) 817-833. | Zbl 1008.76046
[32] , A family of mixed finite elements for the elasticity problem. Numer. Math. 53 (1988) 513-538. | Zbl 0632.73063
[33] , Stabilized hybrid finite element methods based on the combination of saddle point principles of elasticity problems. Math. Comput. 72 (2003) 1655-1673. | Zbl 1081.74046
[34] and, Analysis of locally stabilized mixed finite element methods for the linear elasticity problem. Chinese J. Engrg Math. 12 (1995) 1-6. | Zbl 0924.73243