Superconvergence And Recovery Techniques Based Efficient Adaptive Finite Element Method

The adaptive finite element method(AFEM)is one of the most effective methods for solving numerical partial differential equations,and is widely used in numerical simulation of scientific and engineering problems.The two key steps of AFEM are a posterior error estimation and mesh adaption.A theoretically mature AFEM is generally based on a residual-type posterior error estimation and bisection refinement.In practical calculation,non-asymptotic accuracy of a posterior error estimation and the excessive iteration steps account for the inefficiency of AFEM.In this thesis,an efficient AFEM is developed for second-order elliptic partial differential equations.On the one hand,a posterior error estimation based on superconvergence and reconstruction techniques is used,which can improve the accuracy of the error estimator;on the other hand,mesh refinement and optimization is used in the process of mesh adaption to ensure the high quality of the adaptive meshes.The combination of superconvergence and mesh adaptive ensures high efficiency of the adaptive loops: the high-quality mesh ensures the high accuracy of finite element solutions and supports the superconvergence of recovery techniques;the a posterior error estimation based on gradient recovery ensures asymptotic exactness of the error estimator;the asymptotic accuracy of a posterior error estimation ensures the effectiveness of the mesh adaption;mesh optimization ensures the high quality of the adaptive mesh.The thesis also designed a new mesh adaption strategy and adaptive algorithm,which can control the adaptive iteration within 7 steps.Several numerical examples are presented to show the effectiveness of the new AFEM algorithm.