An anisotropic Zienkiewicz–Zhu‐type error estimator for 3D applications

TLDR

The study extends the anisotropic Zienkiewicz–Zhu a posteriori error estimator to three dimensions. The extended estimator remains problem‑independent, inexpensive, and easy to implement, incorporates the geometry of each tetrahedron, and employs a metric‑based optimization procedure to drive mesh adaptation. Numerical experiments show that the estimator detects anisotropic solution features, yields quasi‑optimal triangulations that balance accuracy and element count, and that the overall adaptation procedure is effective despite its heuristic aspects. © 2010 John Wiley & Sons, Ltd.

Abstract

Abstract We extend the anisotropic Zienkiewicz–Zhu a posteriori error estimator of ( Proceedings of the ENUMATH‐2009 , Uppsala, Sweden, 29 June–3 July 2009) 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. Copyright © 2010 John Wiley & Sons, Ltd.