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Global Superconvergence and A Posteriori Error Estimates of Finite Element Method for Second-order Quasilinear Elliptic Problems

Abstract. In this paper, we are concerned with the linear finite element approximations to the second-order quasi-linear elliptic problems. By means of an interpolation postprocessing technique, we develop the global superconvergence estimates in the $H^1$- and $W^{1,\infty}$-norms provided the weak solutions are sufficiently smooth. Based on the global superconvergent approximations, we introduce and analyze the efficient postprocessing-based a posteriori error estimators, measured by the $H^1$- and $W^{1,\infty}$-norms respectively. These can be used to assess the accuracy of the finite element solutions in applications. Numerical experiments are given to illustrate the global superconvergence estimates and the performance of the proposed estimators.