Analysis of some quadrilateral nonconforming elements for incompressible elasticity

SIAM Journal on Numerical Analysis

Volumn 34, Issue 2, 1997, Pages 640-663

  Zhang, Zhimin ^a  

^a TEXAS TECH UNIVERSITY   (United States)

Author keywords

Incompressible elasticity; Mixed methods; Nonconforming methods

Indexed keywords

EID: 0347981194     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142995282492     Document Type: Article

References (27)

1. D. N. ARNOLD, On nonconforming linear-constant elements for some variants of the Stokes equations, Istit. Lombardo Accad. Sci. Lett. Rend. A, 127 (1993), pp. 83-93.
2. D. N. ARNOLD AND F. BREZZI, Mixed and nonconforming finite element methods: Implementation, post-processing, and error estimates, RAIRO Modél. Math. Anal. Numér., 19 (1985), pp. 7-32.
3. K. ARUNAKIRINATHAR AND B. D. REDDY, Further results for enhanced strain methods with isoparametric elements, Comput. Methods Appl. Mech. Engrg., 127 (1995).
4. S. C. BRENNER AND L.-Y. SUNG, Linear finite element methods for planar linear elasticity, Math. Comp., 59 (1992), pp. 321-338.
5. F. BREZZI AND M. FORTIN, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.
6. I. BABUŠKA AND M. SURI, On locking and robustness in the finite element method, SIAM J. Numer. Anal., 29 (1992), pp. 1261-1293.
7. H. CHEN AND B. LI, Superconvergence analysis and error expansion for the Wilson noncon- forming finite element, Numer. Math., 69 (1994), pp. 125-140.
8. M. CROUZEIX AND P.-A. RAVIART, Conforming and nonconforming finite element methods for solving the stationary Stokes equations, RAIRO Anal. Numèr., 7 R-3 (1973), pp. 33-76.
9. R. S. FALK, Nonconforming finite element methods for the equations of linear elasticity, Math. Comp., 57 (1991), pp. 529-550.
10. M. FROIER, L. NILSSON, AND A. SAMUELSON, The rectangular plane stress element by Turner, Pian and Wilson, Internat. J. Numer. Methods Engrg., 7 (1973), pp. 433-473.
11. V. GlRAULT AND P. A. RAVIART, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer-Verlag, New York, 1986.
12. P. LESAINT, On the convergence of Wilson's nonconforming element for solving the elastic problems, Comput. Methods Appl. Mech. Engrg., 7 (1976), pp. 1-16.
13. P. LESAINT AND M. ZLÁMAL, Convergence of the nonconforming Wilson element for arbitrary quadrilateral meshes, Numer. Math., 36 (1980), pp. 33-52.
14. T. H. H. PIAN, Derivation of element stiffness matrices by assumed stress distributions, AIAA J., 2 (1964), pp. 1333-1336.
15. T. H. H. PIAN AND K. SUMIHARA, Rational approach for assumed stress finite element methods, Internat. J. Numer. Methods Engrg., 20 (1985), pp. 1685-1695.
16. J. PITKÀRANTA AND R. STENBERG, Error bounds for the approximation of the Stokes problem using bilinear/constant elements on irregular quadrilateral meshes, in The Mathematics of Finite Elements and Applications V, J. R. Whiteman, ed., Academic Press, London, 1985, pp. 325-334.
17. B. D. REDDY AND J. C. SIMO, Stability and convergence of a class of enhanced strain methods, SIAM J. Numer. Anal., 32 (1995), pp. 1705-1728.
18. Z. C. SHI, A convergence condition for the quadrilateral Wilson element, Numer. Math., 44 (1984), pp. 349-361.
19. Z. C. SHI, A remark on the optimal order of convergence of Wilson nonconforming element, Math. Numer. Sinica, 8(2) (1986), pp. 159-163. (In Chinese.)
20. J. C. SIMO AND M. S. RIFAI, A class of mixed assumed strain methods and the method of incompatible modes, Internat. J. Numer. Methods Engrg., 29 (1990), pp. 1595-1638.
21. R. STENBERG, Analysis of mixed finite element methods for the Stokes problem: A unified approach, Math. Comp., 42 (1984), pp. 9-23.
22. R. STENBERG AND M. SURI, Mixed hp finite element methods for problems in elasticity and Stokes flow, Numer. Math., 72 (1996), pp. 367-390.
23. F. STUMMEL, The generalized patch test, SIAM J. Numer. Anal., 16 (1979), pp. 449-471.
24. R. L. TAYLOR, P. J. BERESFORD, AND E. L. WILSON, A nonconforming element for stress analysis, Internat. J. Numer. Methods. Engrg., 10 (1976), pp. 1211-1219.
25. E. L. WILSON, R. L. TAYLOR, W. DOHERTY, AND J. GHABOUSSI, Incompatible displacement modes, in Numerical and Computer Models in Structural Mechanics, S. J. Fenves, et al., eds., Academic Press, New York, 1973.
26. Z. ZHANG, Stability Analysis of Wilson's Element for Incompressible Elasticity, Tech, report no. 94-ZZ2, Department of Mathematics, Texas Tech University, Lubbock, TX, June, 1994.
27. Z. ZHANG AND S. ZHANG, Wilson's element for the Reissner-Mindlin plate, Comput. Methods Appl. Mech. Engrg., 113 (1994), pp. 55-65.