3.1.6  Final Remarks on FEM

The Finite Element Method is a vast and still very active subject of research. This text is not intended to discuss it exhaustively, but to give a first intuitive approach and still formal enough to understand this work. However, some points are still open, for instance, an understanding of the conditions under which the FEM converges is still needed. As mentioned previously, one expects that the discretized solution (uh  ) converges to the exact solution (u  ) by increasing the Vh  space dimension – in other words, increasing the number of mesh points. Actually, this discussion leads to restrictions of the basis functions to ensure FEM convergence (and also stability) [56][59]. An in-depth treatment about this topic can be found in specific functional analysis texts, as well as in FEM books [56][57][59].

Another point of contention is the boundary conditions. In the previous sections, only specific Dirichlet boundary conditions [61] were discussed. Indeed, for general Dirichlet conditions all the previous results remain unchanged, less a constant, but for Neumann boundary conditions [61] a new development is needed, especially for the equivalence between the BVP problem and the variational formulation. The procedure for deriving FEM is still the same, as well as the conclusions, but the arguments must be modified to include this kind of situation. A self-contained text about this subject can be found in traditional FEM references [56]; in addition, an example with a Neumann boundary condition will be provided in the next section.