Analysis of a new augmented mixed finite element method for linear elasticity allowing ℝ𝕋 0 - 1 - 0 approximations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 1, p. 1-28
We present a new stabilized mixed finite element method for the linear elasticity problem in 2 . 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.
DOI : https://doi.org/10.1051/m2an:2006003
Classification:  65N12,  65N15,  65N30,  74B05
Keywords: mixed-FEM, augmented formulation, linear elasticity, locking-free
@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},
     publisher = {EDP-Sciences},
     volume = {40},
     number = {1},
     year = {2006},
     pages = {1-28},
     doi = {10.1051/m2an:2006003},
     zbl = {pre05038390},
     mrnumber = {2223502},
     language = {en},
     url = {http://www.numdam.org/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, Volume 40 (2006) no. 1, pp. 1-28. doi : 10.1051/m2an:2006003. http://www.numdam.org/item/M2AN_2006__40_1_1_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.N. Arnold, F. Brezzi and M. Fortin, A stable finite element method for the Stokes equations. Calcolo 21 (1984) 337-344. | Zbl 0593.76039

[3] D.N. Arnold, J. Douglas and Ch.P. Gupta, A family of higher order mixed finite element methods for plane elasticity. Numer. Math. 45 (1984) 1-22. | Zbl 0558.73066

[4] D. Arnold and R. Falk, 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] I. Babuška and G.N. Gatica, On the mixed finite element method with Lagrange multipliers. Numer. Methods Partial Differential Equations 19 (2003) 192-210. | Zbl 1021.65056

[6] M. Barrientos, G.N. Gatica and E.P. Stephan, 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] T.P. Barrios, G.N. Gatica and F. Paiva, A wavelet-based stabilization of the mixed finite element method with Lagrange multipliers. Appl. Math. Lett. (in press). | MR 2202412 | Zbl 1101.65106

[8] D. Braess, Finite Elements. Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press (1997). | MR 1463151 | Zbl 0894.65054

[9] F. Brezzi and J. Douglas, Stabilized mixed methods for the Stokes problem. Numer. Math. 53 (1988) 225-235. | Zbl 0669.76052

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

[11] F. Brezzi and M. Fortin, A minimal stabilisation procedure for mixed finite element methods. Numer. Math. 89 (2001) 457-491. | Zbl 1009.65067

[12] F. Brezzi, J. Douglas and L.D. Marini, Variable degree mixed methods for second order elliptic problems. Mat. Apl. Comput. 4 (1985) 19-34. | Zbl 0592.65073

[13] F. Brezzi, J. Douglas and L.D. Marini, Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47 (1985) 217-235. | Zbl 0599.65072

[14] F. Brezzi, M. Fortin and L.D. Marini, Mixed finite element methods with continuous stresses. Math. Models Methods Appl. Sci. 3 (1993) 275-287. | Zbl 0774.73066

[15] D. Chapelle and R. Stenberg, 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] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland (1978). | MR 520174 | Zbl 0383.65058

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

[18] H.-Y. Duan and G.-P. Liang, 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] L.P. Franca, New Mixed Finite Element Methods. Ph.D. Thesis, Stanford University (1987).

[20] L.P. Franca and T.J.R. Hughes, Two classes of finite element methods. Comput. Methods Appl. Mech. Engrg. 69 (1988) 89-129. | Zbl 0629.73053

[21] L.P. Franca and A. Russo, Unlocking with residual-free bubbles. Comput. Methods Appl. Mech. Engrg. 142 (1997) 361-364. | Zbl 0890.73064

[22] L.P. Franca and R. Stenberg, Error analysis of Galerkin least squares methods for the elasticity equations. SIAM J. Numer. Anal. 28 (1991) 1680-1697. | Zbl 0759.73055

[23] C.O. Horgan, Korn's inequalities and their applications in continuum mechanics. SIAM Rev. 37 (1995) 491-511. | Zbl 0840.73010

[24] C.O. Horgan and J.K. Knowles, Eigenvalue problems associated with Korn's inequalities. Arch. Rational Mech. Anal. 40 (1971) 384-402. | Zbl 0223.73011

[25] C.O. Horgan and L.E. Payne, On inequalities of Korn, Friedrichs and Babuška-Aziz. Arch. Rational Mech. Analysis 82 (1983) 165-179. | Zbl 0512.73017

[26] N. Kechkar and D. Silvester, Analysis of locally stabilized mixed finite element methods for the Stokes problem. Math. Comput. 58 (1992) 1-10. | Zbl 0738.76040

[27] G. Lube and A. Auge, Stabilized mixed finite element approximations of incompressible flow problems. Zeitschrift für Angewandte Mathematik und Mechanik 72 (1992) T483-T486. | Zbl 0766.76050

[28] W. Mclean, Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press (2000). | MR 1742312 | Zbl 0948.35001

[29] A. Masud and T.J.R. Hughes, A stabilized mixed finite element method for Darcy flow. Comput. Methods Appl. Mech. Engrg. 191 (2002) 4341-4370. | Zbl 1015.76047

[30] R. Nascimbene and P. Venini, 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] S. Norburn and D. Silvester, Fourier analysis of stabilized Q 1 -Q 1 mixed finite element approximation. SIAM J. Numer. Anal. 39 (2001) 817-833. | Zbl 1008.76046

[32] R. Stenberg, A family of mixed finite elements for the elasticity problem. Numer. Math. 53 (1988) 513-538. | Zbl 0632.73063

[33] T. Zhou, 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] T. Zhou and L. Zhou, Analysis of locally stabilized mixed finite element methods for the linear elasticity problem. Chinese J. Engrg Math. 12 (1995) 1-6. | Zbl 0924.73243