  Compared to conforming P1 finite elements, nonconforming P1 finite element
discretizations are thought to be less sensitive to the appearance of distorted
triangulations. E.g., optimal-order discrete $H^1$ norm best approximation
error estimates for $H^2$ functions hold for arbitrary triangulations. However,
similar estimates for the error of the Galerkin projection for second-order
elliptic problems show a dependence on the maximum angle of all triangles in
the triangulation. We demonstrate on the example of a special family of
distorted triangulations that this dependence is essential, and due to the
deterioration of the consistency error. We also provide examples of sequences
of triangulations such that the nonconforming P1 Galerkin projections for a
Poisson problem with polynomial solution do not converge or converge at
arbitrarily slow speed. The results complement analogous findings for
conforming P1 elements.
