Analysis of a new augmented mixed finite element method for linear elasticity allowing ℝ double-struck T sign0-ℙ1- ℙ0 approximations

Mathematical Modelling and Numerical Analysis

Volumn 40, Issue 1, 2006, Pages 1-28

  Gatica, Gabriel N ^a  

^a UNIVERSITY OF CONCEPCIÓN   (Chile)

Author keywords

Augmented formulation; Linear elasticity; Locking free; Mixed FEM

Indexed keywords

EID: 33646681922     PISSN: 0764583X     EISSN: 12903841     Source Type: Journal    
DOI: 10.1051/m2an:2006003     Document Type: Article

References (34)

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.
2. D.N. Arnold, F. Brezzi and M. Fortin, A stable finite element method for the Stokes equations. Calcolo 21 (1984) 337-344.
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.
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.
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.
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.
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).
8. D. Braess, Finite Elements. Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press (1997).
9. F. Brezzi and J. Douglas, Stabilized mixed methods for the Stokes problem. Numer. Math. 53 (1988) 225-235.
10. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag (1991).
11. F. Brezzi and M. Fortin, A minimal stabilisation procedure for mixed finite element methods, Numer. Math. 89 (2001) 457-491.
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.
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.
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.
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.
16. P.G. Ciarlet, The Finite Element Method J or Elliptic Problems. North-Holland (1978).
17. J. Douglas and J. Wan, An absolutely stabilized finite element method for the Stokes problem. Math. Comput. 52 (1989) 495-508.
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.
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.
21. L.P. Franca and A. Russo, Unlocking with residual-free bubbles. Comput. Methods Appl. Mech. Engrg. 142 (1997) 361-364.
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.
23. C.O. Horgan, Korn's inequalities and their applications in continuum mechanics. SIAM Rev. 37 (1995) 491-511.
24. C.O. Horgan and J.K. Knowles, Eigenvalue problems associated with Korn's inequalities. Arch. Rational Mech. Anal. 40 (1971) 384-402.
25. C.O. Horgan and L.E. Payne, On inequalities of Korn, Friedrichs and Babuška-Aziz. Arch. Rational Mech. Analysis 82 (1983) 165-179.
26. N. Kechkar and D. Silvester, Analysis of locally stabilized mixed finite element methods for the Stokes problem. Math. Comput. 58 (1992) 1-10.
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.
28. W. McLean, Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press (2000).
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.
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.
31. 1 mixed finite element approximation. SIAM J. Numer. Anal. 39 (2001) 817-833.
32. R. Stenberg, A family of mixed finite elements for the elasticity problem. Numer. Math. 53 (1988) 513-538.
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.
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.