1.4 The Reference Elements

In the previous sections, we presented two applications of the finite element method. We have seen that this method is based on a meshing of the used domain for the definition, per piece, of the shape functions whose adequate weighting ensures the approximation of the solution. In the first 1D example, we have used segments which are the only finite elements available in this space. In the second example, we have used triangles, but we could also have meshed the domain in quadrangles. Triangles are often used in 2D, because they are suitable for an automatic meshing. However, quadrangles are appreciated, because they are particularly appropriate for the discretization of narrow zones or involving particular physical phenomena. In addition, it is possible to mix, within the same meshing, different types of elements, provided the continuity conditions necessary for the functions of approximation are fulfilled. The developments carried out in two dimensions are extended in an obvious way to three dimensions. In this case, the space would have been discretized in elements of tetrahedral, hexahedral, prismatic or pyramidal volume.

We thus have several topologies of 1D, 2D and 3D elements. For each topology, we can exploit the quality of the approximation. In the 1D example, we have successively used two polynomial approximations of first order and second order. In the 2D example, we have implemented triangles with first-order polynomial interpolation. We could also have used second order or a higher order or have mixed the orders by...