ABSTRACT
Finite element methods have proved to be much more powerful than finite difference methods for convection-diffusion problems, both practically and theoretically; and the finite volume methods are also best treated as variants of finite element methods. This chapter describes standard finite element methods that use the same space of functions for both approximate solutions and weighting functions, which we refer to as Galerkin methods. A feature of finite element methods that distinguishes them from finite differences is that in their derivation and implementation attention is focused on the computations in an individual element. A piecewise linear approximation on a triangle requires three parameters for its definition, and two of these need to be shared with a neighbouring triangle in order to ensure continuity and therefore that the approximation is conforming.
