Summary: MIXED FINITE ELEMENT METHODS FOR ELLIPTIC PROBLEMS*
DOUGLAS N. ARNOLD
Abstract. 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 rŽesumŽe 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.
Key words. mixed method, finite element, variational principle
1. Introduction. The term mixed method was first used in the 1960's to describe
finite element methods in which both stress and displacement fields are approximated as
primary variables. We begin with the most classical example, the system of linear elasticity.
The equations of linear elasticity consist of the constitutive equation
AS = E(u) in
and the equilibrium equation
div S = f in .
Here denotes the region in three dimensional space, R3