Mixed Finite Element Methods

Finite element methods in which two spaces are used to approximate two different variables receive the general denomination of mixed methods. In some cases, the second variable is introduced in the formulation of the problem because of its physical interest and it is usually related with some derivatives of the original variable. This is the case, for example, in the elasticity equations, where the stress can be introduced to be approximated at the same time as the displacement. In other cases there are two natural independent variables and so, the mixed formulation is the natural one. This is the case of the Stokes equations, where the two variables are the velocity and the pressure. The mathematical analysis and applications of mixed finite element methods have been widely developed since the seventies. A general analysis for this kind of methods was first developed by Brezzi [13]. We also have to mention the papers by Babuska [9] and by Crouzeix and Raviart [22] which, although for particular problems, introduced some of the fundamental ideas for the analysis of mixed methods. We also refer the reader to [32, 31], where general results were obtained, and to the books [17, 45, 37]. The rest of this work is organized as follows: in Section 2 we review some basic tools for the analysis of finite element methods. Section 3 deals with the mixed formulation of second order elliptic problems and their finite element approximation. We introduce the Raviart-Thomas spaces [44, 49, 41] and their generalization to higher dimensions, prove some of their basic properties, and construct the Raviart-Thomas interpolation operator which is a basic tool for the analysis of mixed methods. Then, we prove optimal order error estimates and a superconvergence result for the scalar variable.