# Finite Element Method (FEM)

## The approximation functions

For the finite element approximation Te(x,y) of T(x,y) to converge against the true solution for the element , the approximation functions have to match a few criteria:

 (1) They have to be continuous according to the weak form of the problem, meaning that all terms of the weak form have to be unequal to zero. Consequently the approximation functions are sufficiently differentiable. (2) The polynomial that make up the approximation function have to be complete, i.e. all terms of the polynomial starting with the constant to the term of the highest degree have to be explicit (not equal to zero). (3) The individual terms of the polynomials have to be linearly independent.
Rectangular element with linear approximation functions.

In the following a simple example shall show the determination of linear approximation functions for a rectangular element.

A simple rectangle is chosen for the elements because the approximation functions are functions of the geometry of the element.

The sides of the elements in direction of the x and y-axis have the length and b respectively.

Additionally, a local coordinate system is defined that has its origin in the vertex 1 (see figure to the right).

The introduction of the local coordinate system is a first taste of the coordinate transformation and the so-called master element that will be explained in chapter 3.1.6.