Goal Oriented, Anisotropic, A Posteriori Error Estimates for the Laplace Equation

Abstract

A posteriori error estimates are presented for the Laplace equation and meshes with large aspect ratio. Error estimates are presented in the natural H1 semi- norm or in the framework of goal oriented error control. The proposed estimator re- lies on anisotropic interpolation estimates derived by Formaggia and Perotto [19, 20] and on Zienckiewicz-Zhu [37, 36] post-processing techniques, thus avoiding the use of numerical techniques in order to compute approximations of the Hessian of the solution.All the constant involved in the error estimates are independent of themesh size and aspect ratio, which enables the use of anisotropic, adaptive finite elements.