We consider a posteriori error estimators that can be applied to anisotropic tetrahedral finite element meshes, i.e. meshes where the aspect ratio of the elements can be arbitrarily large. Two kinds of Zienkiewicz-Zhu (ZZ) type error estimators are derived which are both based on some recovered gradient. Two different, rigorous analytical approaches yield the equivalence of both ZZ error estimators to a known residual error estimator. Thus reliability and efficiency of the ZZ error estimation is obtained. Particular attention is paid to the requirements on the anisotropic mesh. The analysis is complemented and confirmed by several numerical examples.

We consider a posteriori error estimators that can be applied to anisotropic tetrahedral finite element meshes, i.e. meshes where the aspect ratio of the elements can be arbitrarily large. Two kinds of Zienkiewicz-Zhu (ZZ) type error estimators are derived which are both based on some recovered gradient. Two different, rigorous analytical approaches yield the equivalence of both ZZ error estimators to a known residual error estimator. Thus reliability and efficiency of the ZZ error estimation is obtained. Particular attention is paid to the requirements on the anisotropic mesh. The analysis is complemented and confirmed by several numerical examples.