When does the finite element method (not) converge?
Various conditions for O(h) convergence of the piecewise linear conforming finite element method (FEM) have been derived in the literature, perhaps the best known being the minimum and maximum angle
conditions derived from estimates for Lagrange interpolation and Cea's lemma. Although these conditions are sufficient, it can be easily shown that they are not necessary, in fact the FEM triangulations can contain
an arbitrary number of arbitrarily 'bad' elements while still exhibiting O(h) convergence in the H^1 (semi)norm. In other words, while the Lagrange interpolation error goes to infinity, the FEM error goes to zero. In this work we generalize the known sufficient conditions for convergence of various orders by modifying the Lagrange interpolation on degenerating elements. Moreover, we derive the first nontrivial necessary condition for various types of FEM convergence. This allows us to construct simple counterexamples, systems of triangulations where the FEM does not have O(h) convergence or does not converge at all. Up to now, only one such counterexample was known in the literature. Although a necessary and sufficient condition remains unknown, the gap between the derived conditions is small in special cases, giving hope for future work.