Three anisotropic benchmark problems for adaptive finite element methods

In this paper we provide three benchmark problems with known exact solutions that can be used to assess the ability of adaptive finite element algorithms to handle anisotropically-behaved solutions. The first one is a Poisson equation with a smooth solution that only changes in one spatial direction. The second one is a singularly-perturbed linear elliptic equation whose solution exhibits a boundary layer, and the last one is a two-equation system that contains a boundary layer in one solution component only. In an appendix we show sample results obtained with the open source library Hermes. (http://hpfem.org/hermes.)