Mixed Finite Element Methods

Abstract

Within the well-known and highly effective finite element method for the computation of approximate solutions of complex boundary value problems, we focus on the often-called mixed finite element methods, where in our terminology the word “mixed” indicates the fact that the problem discretization typically results in a linear algebraic system characterized by a null matrix on the main diagonal. Accordingly, the goals of the present article are (i) to sketch out that several physical problems share such an algebraic structure once a discretization is introduced; (ii) to present a simple, algebraic version of the abstract theory that rules most applications of mixed finite element methods; (iii) to give several examples of efficient mixed finite element methods; and (iv) finally, to give some hints on how to perform a stability and error analysis, focusing on a representative problem.