Flux Recovery and A Posteriori Error Estimators: Conforming Elements for Scalar Elliptic Equations

Abstract

In this paper, we first study two flux recovery procedures for the conforming finite element approximation to general second-order elliptic partial differential equations. One is accurate in a weighted $L^2$ norm studied in [Z. Cai and S. Zhang, SIAM J. Numer. Anal., 47 (2009), pp. 2132–2156] for linear elements, and the other is accurate in a weighted $H(\mathrm{div})$ norm, up to the accuracy of the current finite element approximation. For the $L^2$ recovered flux, we introduce and analyze an a posteriori error estimator that is more accurate than the explicit residual-based estimator. Based on the $H(\mathrm{div})$ recovered flux, we introduce two a posteriori error estimators. One estimator may be regarded as an extension of the recovery-based estimator studied in [Z. Cai and S. Zhang, SIAM J. Numer. Anal., 47 (2009), pp. 2132–2156] to higher-order conforming elements. The global reliability and the local efficiency bounds for this estimator are established provided that the underlying problem is neither convection- nor reaction-dominant. The other is proved to be exact locally and globally on any given mesh with no regularity assumptions with respect to a norm depending on the underlying problem. Numerical results on test problems for these estimators are also presented.