An anisotropic Zienkiewicz-Zhu a posteriori error estimator for 3D applications
Wednesday 26th August 2009
Farrell, P.E.; Micheletti, Stefano; Perotto, Simona
We extend the anisotropic Zienkiewicz-Zhu a posteriori error estimator of  to three dimensions. Like the standard Zienkiewicz-Zhu estimator, the proposed estimator is designed to be independent of the problem at hand, is cheap to compute and easy to implement. In contrast to the standard Zienkiewicz-Zhu estimator, the elementwise counterpart of the proposed estimator explicitly takes into account the geometrical properties of the actual tetrahedron. Thus, in a wide variety of applications, the estimator is able to detect the anisotropic features exhibited by the solution of the governing equations. A metric-based optimization procedure, rigorously addressed, drives the adaptation of the mesh. It is shown numerically to yield quasi-optimal triangulations, dictating the accuracy-vs-number of elements behaviour. Despite being heuristic to some extent, in practice the overall anisotropic adaptation procedure turns out to be effective.