Goal Oriented A Posteriori Error Estimation And Convergent Adaptive Finite Element Method For Elliptic Equation

In this paper,we consider the goal-oriented a posteriori error estimations and analyze the convergence of the corresponding adaptive finite element method for the second order elliptic equation.The error estimations are based on the gradient or flux recovery methods.In the gradient recovery based error estimation,we de-fine a new error estimator and prove its reliability and efficiency.The convergence analysis of the corresponding adaptive finite element is also presented.For the flux recovery based error estimation,we recover the numerical flux into the subspace of H(div)space through the weighted averaging or L2 projection.The corresponding error estimator is reliable and efficient,and the adaptive finite element method is con-vergent.In the end,we provide some numerical examples to confirm the correctness of the theoretical results.