How accurate are finite elements on anisotropic triangulations in the maximum norm?
For researchers using finite elements on anisotropic meshes, this work reveals unexpected convergence behavior that challenges standard assumptions, though it is incremental as it extends known counterexamples.
The paper investigates pointwise convergence of Lagrange finite elements on anisotropic triangulations, showing that linear and higher-order elements can exhibit lower than expected convergence rates for the Laplace equation and singular equations, with accuracy depending on mesh topology beyond interpolation error.
In [Kopteva, Math. Comp., 2014] a counterexample of an anisotropic triangulation was given on which the exact solution has a second-order error of linear interpolation, while the computed solution obtained using linear finite elements is only first-order pointwise accurate. This example was given in the context of a singularly perturbed reaction-diffusion equation. In this paper, we present further examples of unanticipated pointwise convergence behaviour of Lagrange finite elements on anisotropic triangulations. In particular, we show that linear finite elements may exhibit lower than expected orders of convergence for the Laplace equation, as well as for certain singular equations, and their accuracy depends not only on the linear interpolation error, but also on the mesh topology. Furthermore, we demonstrate that pointwise convergence rates which are worse than one might expect are also observed when higher-order finite elements are employed on anisotropic meshes. A theoretical justification will be given for some of the observed numerical phenomena.