# The Finite Element Method

The method consists in aproximating the function in small domain portions called finite elements, or simplex. These elements can be different of quadrilaterals, as it happens with Finite Differences. They can be triangles, quadrilaterals with four or eight nodes, etc.
The unknown function u is approximated in every element by an interpolating polynomial that is continuous with its derivative until a certain order. Once the approximation in each element is determined, the solution for the global domain is obtained assembling the solution found at every element.
To obtain the interpolating polynomials, this method uses the shape functions (N(x,y)). There is a shape function for every simplex node. Ni(x,y) is equal to one in the i node and zero in the others. You can observe the shape functions for the one dimensional element with 2 and three nodes.

 N1,1(x) N2,1(x)
 N1,2(x) N2,2(x) N3,2(x)

To solve a problem by means of the FE method, we have to follow the following basic steps:

• Establishment of stiffness relations for each element. Material properties and equilibrium conditions for each element are used in this establishment.

• Enforcement of compatibility, i.e. the elements are connected. \item Enforcement of equilibrium conditions for the whole structure, in the present case for the nodal points.

• By means of the two previous steps, the system of equations is constructed for the whole structure. This step is called assembling.

• In order to solve the system of equations for the whole structure, the boundary conditions are enforced.

• Solution of the system of equations.