A Posteriori Error Estimates of Finite Element Method for A Quasi-Linear Elliptic Problem of Nonmonotone Type

Abstract. In this paper, we introduce and analyze the residual-type and postprocessing-type a posteriori error estimators of the finite element method for a quasi-linear elliptic problem of nonmonotone type. For the residual-type a posteriori error estimators, we derive the computable upper and lower bounds on the error in the H¹-norm. Based on the global superconvergent approximations of the finite element method, we provide efficient postprocessing-type a posteriori error estimators, measured by the H¹- and W^1,∞-norms respectively. These can be used to assess the accuracy of the finite element solutions in applications. Numerical experiments are given to illustrate the performance of the proposed estimators.

Keywords. quasi-linear elliptic problems, finite element method, a posteriori error estimates, residual estimators, postprocessing estimators, superconvergence