Complements on Mixed Methods for Elliptic Problems

Abstract

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.