Divergence of fem: Babuška-Aziz triangulations revisited

By re-examining the arguments and counterexamples in I. Babuska, A. K. Aziz (1976) concerning the well-known maximum angle condition, we study the convergence behavior of the linear finite element method (FEM) on a family of distorted triangulations of the unit square originally introduced by H. Schwarz in 1880. For a Poisson problem with polynomial solution, we demonstrate arbitrarily slow convergence as well as failure of convergence if the distortion of the triangulations grows sufficiently fast. This seems to be the first formal proof of divergence of the FEM for a standard elliptic problem with smooth solution.