Various Examples

Abstract

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.