Anisotropic Mesh Construction and Error Estimation in the Finite Element Method
Gerd Kunert (TU Chemnitz)
In an anisotropic adaptive finite element algorithm one usually needs an
error estimator that yields the error size but also the stretching
directions and stretching ratios of the elements of a (quasi) optimal
anisotropic mesh. However, the last two ingredients can not be extracted
from any of the known anisotropic a posteriori error estimators.
Therefore a heuristic approach is pursued here, namely, the desired
information is provided by the so-called Hessian strategy. This strategy
produces favourable anisotropic meshes which result in a small
The focus of this paper is on error estimation on anisotropic meshes.
It is known that such error estimation is reliable and efficient only
if the anisotropic mesh is aligned with the anisotropic solution.
The main result here is that the Hessian strategy produces anisotropic
meshes that show the required alignment with the anisotropic solution.
The corresponding inequalities are proven, and the underlying heuristic
assumptions are given in a stringent yet general form. Hence the
analysis provides further insight into a particular aspect of anisotropic