Flux recovery and a posteriori error estimators: Conforming elements for scalar elliptic equations

摘要

In this paper, we.rst study two.ux recovery procedures for the conforming.nite element approximation to general second-order elliptic partial di.erential equations. One is accurate in a weighted L2 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(div) norm, up to the accuracy of the current.nite element approximation. For the L2 recovered.ux, we introduce and analyze an a posteriori error estimator that is more accurate than the explicit residual-based estimator. Based on the H(div) recovered.ux, 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 e.ciency 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.