Numer. Math. 48, 447-462 (1986) Numerische MathemalJk 9 Springer-Verlag 1986 On the Construction of Optimal Mixed Finite Element Methods for the Linear Elasticity Problem Roll Stenberg Institute of Mathematics, Helsinki University of Technology SF-02150 Espoo 15, Finland Summary. The mixed finite element method for the linear elasticity problem is considered. We propose a systematic way of designing methods with optimal convergence rates lor both the stress tensor and the displacement. The ideas are applied in some examples. Subject Classifications: AMS(M OS): 65 N 30; CR: G 1.8. 1. Introduction Recently there has been several attempts to design mixed finite element meth- ods for the linear elasticity problem, i.e. methods where one uses simultaneous approximations for both the stresses and the displacement; cf. [-13, 16, 17]. The motivation for using such a method is to obtain directly an accurate approxi- mation for the stresses; the variables which usually are or primary interest. As is well known, the traditional way is to use a displacement method in which the stresses are obtained through a numerical differentiation of the displace- ments, thus leading to a decrease in accuracy. The mathematical theory of mixed methods is based on the works of Babu~ka [3, 4] and Brezzi [8]. In the elasticity problem the theory has been usually interpreted in such a way that a mixed method has to satisfy two conditions in order to converge, i.e. a "stability condition" and an "equilibrium condition" (cf. Sect. 2 below). For the continuous problem the conditions are valid, but it seems, however, to be very difficult to design simple mixed finite element methods satisfying these conditions. As a consequence, very few meth- ods have been possible to analyze [13, 17]. The methods of [13, 17] are rather complicated and the usefulness of the mixed approach has been questionable despite the fact that some methods can be found which seem to be asymptoti- cally more efficient than the corresponding displacement method with the same order of accuracy; cf. [17]. This paper is a continuation to [17] and the purpose is to point out an alternative way of applying the theory of Babu~ka-Brezzi which seems to be