1.5 Conclusion

We have presented the finite element method on electrostatic examples. This very general method is also applicable to static or dynamic problems of electromagnetism and to multi-physical problems, in particular electromechanical and thermoelectric problems appearing in actuators, sensors and electromechanical devices of all sizes.

In this presentation, we have focused on nodal elements because they are appropriate both to the interpolation of geometry and to the state variable (the electric potential) selected in this chapter.

There are other interpolation possibilities which will be selected according to the characteristics of the fields to interpolate. Thus, the continuous scalar functions, such as the scalar potentials or the temperature, are interpolated naturally with nodal elements. The vector fields, such as electric or magnetic fields, requiring on the interface between two mediums a continuity of their tangential component with a possible discontinuity of their normal component, are processed with edge elements. The vector fields, such as electric or magnetic inductions, with continuity of the normal component and possible discontinuity of their tangential component, are interpolated with facet elements. Lastly, the scalar functions that are continuous per piece, such as charge densities, are interpolated naturally by the volume elements [BOS 93].

Lastly, for a given element topology and a given mode of interpolation, it is also possible to exploit the order of the elements, either by defining for each order families of different functions (as was done in this chapter for first and second-order nodal elements), or by adding to each rise in...