Mixed Finite Element Methods
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 Babuška [9] an
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Introduction 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 Babuˇska [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 Sect. 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. We follow the ideas developed in several papers (see for example [24, 16]). Although for simplicity we consider the Raviart–Thomas spaces, the error analysis depends only on some basic properties of the spaces and the interpolation operator, and therefore, analogous results hold for approximations obtained with other finite element spaces. We end the section recalling other known families of spaces and giving some references. In Sect. 4 we introduce an a posteriori
2
R.G. Dur´an
error estimator and prove its equivalence with an appropriate norm of the error up to higher order terms. For simplicity, we present the a posteriori error analysis only in the 2-d case. Finally, in Sect. 5, we introduce the general abstract setting for mixed formulations and prove general existence and approximation results.
2 Preliminary Results In this section we recall some basic results for the analysis of finite element approximations. We will use the standard notation for Sobolev spaces and their norms, namely, given a domain Ω ⊂ IRn and any positive integer k H k (Ω) = {φ ∈ L2 (Ω) : Dα φ ∈ L2 (Ω) ∀ |α| ≤ k}, where α = (α1 , · · · , αn ),
|α| = α1 + · · · + αn
and
Dα φ =
∂ |α| φ n · · · ∂xα n
1 ∂xα 1
and the derivatives are taken in the distribu
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