Reliability and efficiency of an anisotropic Zienkiewicz–Zhu error estimator

In this paper we study the efficiency and the reliability of an anisotropic a posteriori error estimator in the case of the Poisson problem supplied with mixed boundary conditions. The error estimator may be classified as a residual-based one, but its novelty is twofold: firstly, it employs anisotropic estimates of the interpolation error for linear triangular finite elements and, secondly, it makes use of the Zienkiewicz–Zhu recovery procedure to approximate the gradient of the exact solution. Finally, we describe the adaptive procedure used to obtain a numerical solution satisfying a given accuracy, and we include some numerical test cases to assess the robustness of the proposed numerical algorithm.