A family of mixed finite elements for the elasticity problem. (English) Zbl 0632.73063

A new mixed finite element formulation for the equations of linear elasticity is considered. In the formulation the variables approximated are the displacement, the unsymmetric stress tensor and the rotation. The rotation act as a Lagrange multiplier introduced in order to enforce the symmetry of the stress tensor. Based on this formulation a new family of both two- and three-dimensional mixed methods is defined. Optimal error estimates, which are valid uniformly with respect to the Poisson ratio, are derived. Finally, a new postprocessing scheme for improving the displacement is introduced and analyzed.


74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65K10 Numerical optimization and variational techniques
Full Text: DOI EuDML


[1] Amara, M., Thomas, J.M.: Equilibrium finite elements for the linear elastic problem. Numer. Math.33, 367-383 (1979) · Zbl 0401.73079
[2] Arnold, D.N., Douglas, J., Gupta, C.P.: A family of higher order mixed finite element methods for plane elasticity. Numer. Math.45, 1-22 (1984) · Zbl 0558.73066
[3] Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Math. Mod. Anal. Numer.19, 7-32 (1985) · Zbl 0567.65078
[4] Arnold, D.N., Brezzi, F., Douglas, J.: PEERS: A new finite element for plane elasticity. Jap. J. Appl. Math.1, 347-367 (1984) · Zbl 0633.73074
[5] Arnold, D.N., Falk, R.S.: Well-posedness of the fundamental boundary value problems for constrained anisotropic elastic materials. Arch. Rat. Mech. Anal.98, 143-167 (1987) · Zbl 0618.73012
[6] Babu?ka, I., Osborn, J., Pitkäranta, J.: Analysis of mixed methods using mesh dependent norms. Math. Comput.35, 1039-1062 (1980) · Zbl 0472.65083
[7] Bercovier, M.: Perturbation of mixed variational problems: Application to mixed finite element methods. RAIRO Anal. Numer.12, 211-236 (1978) · Zbl 0428.65059
[8] Boland, J.M., Nicolaides, R. A.: Stability of finite elements under divergence constraints. SIAM J. Num. Anal.20, 722-731 (1983) · Zbl 0521.76027
[9] Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. RAIRO Ser. Rouge8, 129-151 (1974) · Zbl 0338.90047
[10] Brezzi, F., Douglas, J., Marini, L.D.: Two families of mixed finite element methods for second order elliptic problems. Numer. Math.47, 217-235 (1985) · Zbl 0599.65072
[11] Brezzi, F., Douglas, J., Marini, L.D.: Recent results on mixed finite element methods for second order elliptic problems. In: Vistas in Applied Mathematics. Numerical Analysis, Atmospheric Sciences and Immunology. pp. 25-43. Heidelberg Berlin New York: Springer 1986 · Zbl 0611.65071
[12] Ciarlet, P.G.: The finite element method for elliptic problems.. Amsterdam: North-Holland 1978
[13] Crouzeix, M., Raviart, P.A.: Conforming and nonconforming finite elements for solving the stationary Stokes equations. RAIRO Ser. Rouge7, 33-76 (1973) · Zbl 0302.65087
[14] Fraijs de Veubeke, B.X.: Displacement and equilibrium models in the finite element method. Stress Analysis. (O.C. Zienkiewicz, G. Holister, eds.), pp. 145-197. New York: Wiley 1965
[15] Fraijs de Veubeke, B.X.: Stress function approach. Bournemouth, World Conference in Finite Elements 1975 pp. J.1?J.51
[16] Johnson, C., Mercier, B.: Some equilibrium finite element methods for two-dimensional elasticity problems. Numer. Math.30, 103-116 (1978) · Zbl 0427.73072
[17] Karp, S.N., Karal, F.C.: The elastic field in the neighbouthood of a crack of arbitrary angle. CPAM15, 413-421 (1962) · Zbl 0166.20705
[18] Mansfield, L.: On mixed finite element methods for elliptic equations. Comput. Math. Appl.7, 59-66 (1981) · Zbl 0458.65088
[19] Mirza, F.A., Olson, M.D.: The mixed finite element method in plane elasticity. Int. J. Numer. Methods Eng.15, 273-290 (1980) · Zbl 0426.73069
[20] Nedelec, J.C.: Mixed finite elements inR 3. Numer. Math.35, 315-341 (1980) · Zbl 0419.65069
[21] Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. Proceedings of the Symposium on the mathematical aspects on the finite element method. Lecture Notes in Mathematics No.606, 292-315. Berlin Heidelberg New York: Springer 1977 · Zbl 0362.65089
[22] Pitkäranta, J., Stenberg, R.: Analysis of some mixed finite element methods for plane elasticity equations. Math. Comput.41, 399-423 (1983) · Zbl 0537.73057
[23] Stenberg, R.: Analysis of mixed finite element methods for the Stokes problem: A unified approach. Math. Comput.42, 9-23 (1984) · Zbl 0535.76037
[24] Stenberg, R.: On the construction of optimal mixed finite element methods for the linear elasticity problem. Numer. Math.48, 447-462 (1986) · Zbl 0563.65072
[25] Stenberg, R.: On the postprocessing of mixed equilibrium finite element methods. In: Numerical techniques in continuum mechanics. Proceedings of the Second GAMM-Seminar, Kiel 1986. (W. Hackbusch, K. Witsch, ed.), pp. 102-109. Braunschweig: Vieweg 1987
[26] Williams, M.L.: Stress singularities resulting from various boundary conditions in angular plates in extension. J. Appl. Mech.19, 526-528 (1952)
[27] Zienkiewicz, O.C., Li, X.-K., Nakazawa S.: Iterative solution of mixed problems and the stress recovery procedures. Comm. Appl. Numer. Methods3, 3-9 (1985) · Zbl 0586.73127
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.