Mixed and Hybrid Finite Element Methods

Although we shall not define in this chapter mixed and hybrid (or other nonstandard) finite element methods in a very precise way, we would like to situate them in a sufficiently clear setting. As we shall see, boundaries between different methods are sometimes rather fuzzy. This will not be a real drawback if we nevertheless know how to apply correctly the principles underlying their analysis.

This chapter is in a sense the kernel of the book. It sets a general framework in which mixed and hybrid finite element methods can be studied. Even if some applications will require variations of the general results, these could not be understood without the basic notions introduced here. Our first concern will be existence and uniqueness of solutions. We first consider in Section II.1.1 the simple case of a saddle point problem corresponding to the minimization of a linearly constrained quadratic functional. This case is extended in Section II.1.2 to a more general case. The matter of approximating the solution will then be considered under various (but classical) assumptions. Finally, we shall deal with numerical properties of the discretized problems and practical computational facts.

This chapter will rapidly present various applications of the theory developed in Chapter II. It will give the reader a general idea of the possibilities offered by this theoretical framework. Many of our examples have already been considered in Chapter I. We shall consider here existence and uniqueness proofs, when they can be obtained, in a proper functional setting. Moreover, we shall give examples of discretizations and error estimates. Some of the problems considered here will be presented in a more detailed treatment in future chapters: this will be the place where special cases and exceptions will eventually be discussed; the present analysis is, in principle, restricted to simple and straightforward cases. We shall, therefore, successively consider non-standard methods for Dirichlet’s problem, including hybrid methods. We shall then present approximations of the Stokes problem and of the linear elasticity problems. Fourth-order problems will also be considered either by mixed methods such as the ψ-ω method (CIARLET-RAVIART [C], MERCIER [A]) or à la MIYOSHI [A] or by dual hybrid methods. This list of examples is obviously not exhaustive and many applications have not been treated, in particular, equilibrium methods for which we refer to BREZZI-MARINI-QUARTERONI-RAVIART [A], HLAVACEK [A], HASLINGER-HLAVACEK [A]-[B] and BATOZ-BATHE-HO [A]. Other examples can be found in ROBERTS-THOMAS [A] and the references therein. Other applications and variants of the methods presented can also be found in BATOZ-BATHE-HO [A], KIKUCHI [A], and QUARTERONI [A,B], RANNACHER [A], and SCAPOLLA [A] for fourth-order problems. Time-dependent problems have been treated in QUARTERONI [C] and with a quite difffrent methodology in HUGHES-HULBERT [A]. Finally, let us point out the contribution (e.g., WHEELER-GONZALEZ [A]) of many people working on reservoir modeling to mixed methods.

In this chapter we present some additional results on the application of the mixed finite element method to linear elliptic problems. In particular in Section V.1 we shall discuss some aspects of the numerical techniques that can be used for solving the linear system of equations that one obtains after discretization. The procedure suggested here is essentially due (to our knowledge) to Fraeijs de Veubeke and, as we shall see, involves the introduction of suitable interelement Lagrange multipliers λ. Such a trick has the remarkable effect of reducing the total number of unknowns and leads to solving a linear system for a matrix which is symmetric and positive definite instead of the original indefinite one. A rough analysis of the computational effort that this procedure requires for the various elements is presented in Section V.2. Moreover, as we shall see in Section V.3, the new unknown λ’s that are obtained by such a procedure allow the construction of a new approximation u^*_h of u, depending on λ and u_h, which is usually much closer to u. In a fourth section we sketch miscellaneous results on error estimates in different norms. Section V.5 is dedicated to an example of application to semiconductor devices simulation. Finally, Section V.6 presents, on a very simple problem, some examples of dicretization that do not work, and Section V.7 applications of augmented formulations introduced in Section I.5.

Although the approximation of incompressible flows by finite element methods has grown quite independently of the main stream of mixed and hybrid methods, it was soon recognized that a precise analysis requires the framework of mixed methods. In many cases, one may apply directly the techniques and results of Chapter II. In particular, the elements used are often standard elements or simple variants of standard elements. The specificity of Stokes problem has, however, led to the development of special techniques; we shall present some of them that seem particularly interesting. Throughout this study the main point will be to make a clever choice of elements leading to the satisfaction of the inf-sup condition. This chapter, after a summary of the problem, will present examples of elements and techniques of proof. It will not be possible to analyze fully all elements for which results are known; we shall try to group them by families which can be treated by similar methods. These families will be arbitrary and will overlap in many cases. Besides this presentation of elements, we shall also consider solution techniques by penalty methods and will develop the related problem of almost incompressible elastic materials. We shall consider the equivalence of penalty methods and mixed methods and some questions arising from it.

In this chapter we shall present a few among the many other applications of mixed methods. In the first section we shall describe a mixed method for linear thin plates theory, in the second section we shall discuss some applications of mixed methods to linear elasticity with a particular stress on the nearly incompressible case, and in the third section we shall report some recent results on the discretization of the Mindlin-Reissner formulation for moderately thick plates.