Mixed Finite Element Methods for Elliptic Problems*

This paper treats the basic ideas of mixed finite element methods at an introductory level. Although the viewpoint presented is that of a mathematician, the paper is aimed at practitioners and the mathematical prerequisites are kept to a minimum. A classification of variational principles and of the corresponding weak formulations and Galerkin methods—displacement, equilibrium, and mixed—is given and illustrated through four significant examples. The advantages and disadvantages of mixed methods are discussed. The concepts of convergence, approximability, and stability and their interrelations are developed, and a résumé is given of the stability theory which governs the performance of mixed methods. The paper concludes with a survey of techniques that have been developed for the construction of stable mixed methods and numerous examples of such methods.