  In this paper we derive a necessary condition for finite element method (FEM)
convergence in $H^1(\Omega)$ as well as generalize known sufficient conditions.
We deal with the piecewise linear conforming FEM on triangular meshes for
Poisson's problem in 2D. In the first part, we prove a necessary condition on
the mesh geometry for $O(h^\alpha)$ convergence in the $H^1(\Omega)$-seminorm
with $\alpha\in[0,1]$. We prove that certain structures, bands consisting of
neighboring degenerating elements forming an alternating pattern cannot be too
long with respect to $h$. This is a generalization of the Babu\v ska-Aziz
counterexample and represents the first nontrivial necessary condition for
finite element convergence. Using this condition we construct several
counterexamples to various convergence rates in the FEM. In the second part, we
generalize the maximum angle and circumradius conditions for $O(h^\alpha)$
convergence. We prove that the triangulations can contain many elements
violating these conditions as long as their maximum angle vertexes are
sufficiently far from other degenerating elements or they form clusters of
sufficiently small size. While a necessary and sufficient condition for
$O(h^\alpha)$ convergence in $H^1(\Omega)$ remains unknown, the gap between the
derived conditions is small in special cases.
