Gradient Recovery Based A Posteriori Error Estimation And The Adaptive Finite Element Method For Semilinear Elliptic Equations

In this paper,we consider the gradient recovery-based a posteriori error estimator and adaptive finite element methods for the semi-linear elliptic equations.Firstly,we propose a recovery based a posteriori error estimator for the linear elliptic equations,and then propose an adaptive finite element algorithm and derive its convergence analysis.Secondly,by exploiting the relation between the semilinear elliptic equation and the corresponding linear equation,we derive a gradient recovery based a posteriori error estimator,and prove its reliable and efficiency.The convergence analysis of the adaptive finite element method are also presented.Some numerical examples are given to show the conclusion performance of the proposed a posteriori error estimator and adaptive algorithm.