Fakultat fur Mathematik
TU Chemnitz
Reichenhainer Str. 41
D-09107 Chemnitz
Deutchland
Many physical problems lead to boundary value problems for partial
differential equations, which can be solved with the finite element
method. In order to construct adaptive solution algorithms or to
measure the error one aims at reliable a posteriori error
estimators. Many such estimators are known, as well as their
theoretical foundation.
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 our talk error estimators for anisotropic tetrahedral or
triangular meshes are considered. Such meshes offer a greater
geometrical flexibility.
For the Poisson equation we present a residual error estimator,
an estimator based on a local problem, several Zienkiewicz-Zhu
estimators, and an $ L_2 $ error estimator, respectively.
For a singularly perturbed reaction-diffusion equation a residual
error estimator is given as well.
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.