MATHEMATICS OF COMPUTATION VOLUME 41. NUMBER 164 OCTOBER 19R3. PAGES 399-423 Analysis of Some Mixed Finite Element Methods for Plane Elasticity Equations By J. Pitkäranta and R. Stenberg Abstract. We analyze some mixed finite element methods, based on rectangular elements, for solving the two-dimensional elasticity equations. We prove error estimates for a method proposed by Taylor and Zienkiewicz and for some new variants of the known equilibrium methods. A numerical example is given demonstrating the performance of the various algorithms considered. 1. Introduction. In the numerical solution of problems of continuum mechanics, the stresses are normally of primary interest in the elastic region. It is therefore natural to design the numerical algorithms so that the stresses can be obtained directly without first computing the displacements. Such methods can be derived from the dual variational formulation of the elasticity problem. The corresponding finite element algorithms are usually formulated as mixed methods where both the displacements and the stresses are first approximated, and the displacements are then eliminated from the discrete equations. In many cases the elimination can be rather effectively done using penalty/perturbation techniques or their iterative variants; cf. [3],[11],[12]. The best known finite element methods of the above type are the so-called equilibrium methods, first proposed by Fraejis de Veubeke [17] (cf. also [14], [16], [18]) and analyzed theoretically by Johnson and Mercier [9] (cf. also [8]). In these methods, one uses specific composite elements which allow the equilibrium condi- tion between the stresses and the volume load to be satisfied exactly in the case where the volume load is zero. The main drawback of the equilibrium methods proposed so far is the relatively high number of free parameters as compared with displacement methods of the same order of accuracy. For example, if the composite quadrilateral element of [17], [9] is used on a regular rectangular grid, one has eight degrees of freedom per each interior node of the grid (after the local condensation of three extra degrees of freedom per node, cf. [9]) and the convergence rate 0(h2) for the stresses in L2 [9]. On the other hand, using the displacement method with reduced biquadratic elements (cf. [6]), one has the same convergence rate with six parameters per node, so the displacement method seems superior. It is clear from the above example that the mixed or equilibrium methods should be further developed if they are desired to be competitive with displacement Received August 18, 1982. 1980 Mathematics Subject Classification. Primary 65N30. ©1983 American Mathematical Society 0025-5718/83 $1.00 + $.25 per page 399 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use