The maximum angle condition is not necessary for convergence of the finite element method

We show that the famous maximum angle condition in the finite element analysis is not necessary to achieve the optimal convergence rate when simplicial finite elements are used to solve elliptic problems. This condition is only sufficient. In fact, finite element approximations may converge even though some dihedral angles of simplicial elements tend to π

Nonconforming P1 elements on distorted triangulations: Lower bounds for the discrete energy norm error

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.Comment: 23 pages, 10 figure