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.



1-D Linear Shape Functions




1-D Quadratic Shape Functions

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

Learn more about the FE method.

Theory pages:
Main page
FEM (i)
FEM (ii)
FDM (i)
Example pages:
1-d Heat Equation
2-d steady state Heat Equation
2-d Heat Equation
1-d non diffusive transport equation
1-d diffusive transport equation
2-d Non diffusive transport equation
Mesh generation with OOCSMP
Moving grids
Application pages:
Heating of two beams
Heating of two moving beams
Solving the equation Ut+Uxx+Uxy+Uyx=0
Solving the equation Ut+Uxx+Uxy+Uyx=0 using MGEN
Heat 1d using several outputs
Solving the Heat equation with a CA
Comparing a CA with the FEM
Gordon's sine equation

Other courses
Other pages

Last modified 4/2/2000 by Juan de Lara (, need help for using this courses?.

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