General

We saw in Section 1.3 that finite element equations are obtained by the classical approximation theories such as variational or weighted residual methods. However, there are some basic differences in philosophy between the classical approximation theories and finite element methods. In the finite element methods, the global functional representations of a variable consist of an assembly of local functional representations so that the global boundary conditions can be implemented in local elements by modification of the assembled algebraic equations. The local interpolation (shape, basis, or trial) functions are chosen in such a manner that continuity between adjacent elements is maintained.

The finite element interpolations are characterized by the shape of the finite element and the order of the approximations. In general, the choice of a finite element depends on the geometry of the global domain, the degree of accuracy desired in the solution, the ease of integration over the domain, etc.