Elliptic Problems: Approximation by Galerkin and Collocation Methods

Abstract

This Chapter deals with second order linear elliptic equations. We present the variational formulation of some classical boundary value problems, accounting for several kind of boundary conditions, and derive existence and uniqueness of the solution. Then we approximate these problems by Galerkin and collocation methods, in the framework of both finite element and spectral methods. For each kind of discretization we analyze its stability and convergence properties, as well as its algorithmic aspects.