Finite Element Method for Elliptic PDEs

Abstract

In this chapter, the main tools of the FEM for elliptic equations will be considered. First, we recall the properties of real Hilbert spaces including the Riesz representation theorem. Then we give examples of commonly used Sobolev spaces and formulate the basic trace and extension theorems as well as some norm equivalence relations which will be used in subsequent sections. We collect Green’s formulae for the scalar second order, biharmonic, Lamé and Stokes elliptic operators. The general approximation results for coercive, indefinite and saddle-point variational problems based on the Galerkin, Petrov-Galerkin and discrete saddle-point schemes, respectively, will be presented.