Nonconforming P1 elements on distorted triangulations: Lower bounds for the discrete energy norm error
This work provides essential lower bounds for practitioners using nonconforming P1 elements on distorted meshes, revealing that such elements are not as robust as previously thought.
The paper shows that nonconforming P1 finite elements on distorted triangulations can suffer from severe convergence degradation, including non-convergence or arbitrarily slow convergence for Poisson problems with polynomial solutions, due to consistency error deterioration dependent on the maximum angle of triangles.
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.