COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 82 (1990) 27-57 NORTH-HOLLAND A DISCOURSE ON THE STABILITY CONDITIONS FOR MIXED FINITE ELEMENT FORMULATIONS Franco BREZZI Dipartimento di Meccanica Strutturale and Istituto di Analisi, Numerica del C. N.R., 27100 Pavia, Italy Klaus-Jiirgen BATHE Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. Received 13 February 1990 We discuss the general mathematical conditions for solvability, stability and optimal error bounds of mixed finite element discretizations. Our objective is to present these conditions with relatively simple arguments. We present the conditions for solvability and stability by considering the general coefficient matrix of mixed finite element discretizations, and then deduce the conditions for optimal error bounds for the distance between the finite element solutions and the exact solution of the mathematical problem. To exemplify our presentation we consider the solutions of various example problems. Finally, we also present a numerical test that is useful to identify numerically whether, for the solution of the general Stokes flow problem, a given finite element discretization satisfies the stability and optimal error bound conditions. I. Introduction During the recent years it has been recognized to an increasing extent that the use of mixed finite elements can be of great benefit and may even be necessary to obtain reliable and accurate solutions in certain fields of engineering analysis. Mixed finite elements are currently used with much success in the solution of incompressible fluid flows, and continue to provide great promise for the analysis of solids and structures [1, 2]. Of course, the largest area of finite element applications is still structural analysis and mixed finite elements are, in principle, much suited for use in the analysis of almost incompressible media (for example, for the analysis of rubber-like materials, elasto-plasticity and creep) and the analysis of plates and shells. However, although many mixed finite elements have been proposed over the last two decades in the research literature, it is apparent that mixed finite elements are hardly used in practical structural analysis. The reason why mixed finite elements are not used abundantly in engineering practice is that their predictive behavior is much more difficult to assess than for the conventional and commonly used displacement-based elements. Whereas displacement-based elements, once formulated and shown to work well on certain sets of examples (including the patch tests), can be generally employed, mixed finite elements cannot be recommended for general use unless a deeper analysis and understanding is available. Namely, considering a certain category of 0045-7825/90/$03.50 © 1990- Elsevier Science Publishers B.V. (North-Holland)