Some boundary value problems yield so-called anisotropic solutions (e.g. with boundary layers). Then anisotropic finite element meshes can be advantageous. However, the common error estimators for isotropic meshes fail when applied to anisotropic meshes, or they were not investigated yet.
For rectangular or cuboidal anisotropic meshes a modified error estimator had already been derived. In this paper error estimators for anisotropic tetrahedral or triangular meshes are considered. Such meshes offer a greater geometrical flexibility.
For the Poisson equation we introduce a residual error estimator, an estimator based on a local problem, several Zienkiewicz-Zhu estimators, and an L 2 error estimator, respectively. A corresponding mathematical theory is given. For a singularly perturbed reaction-diffusion equation a residual error estimator is derived as well. The numerical examples demonstrate that reliable and efficient error estimation is possible on anisotropic meshes.
The analysis basically relies on two important tools, namely anisotropic interpolation error estimates and the so-called bubble functions. Moreover, the correspondence of an anisotropic mesh with an anisotropic solution plays a vital role.