Approximation of Saddle Point Problems

Abstract

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.