自适应有限元方法是数值求解科学计算和工程问题的最有效方法之一，有限元超收敛方法和重构技术在后验误差估计及自适应有限元方法中具有很高的应用价值和前景。本项目旨在促进有限元超收敛和重构技术在自适应有限元方法中的应用，深入研究有限元的超收敛和函数值、梯度及通量重构方法，并构造基于各种重构方法的后验误差估计及自适应算法。与已有的研究工作相比较，我们将放松超收敛对网格的强要求，找出可计算的参数来量化网格质量对误差收敛阶的影响，分析CVDT网格上的超收敛性；并根据研究结果设计网格加密与优化算法，使得生成的网格保持超收敛性；我们还提出新的精度高、实现简单且有效的重构算法及相应的后验误差估计；设计基于新的超收敛结果和重构技术的自适应有限元方法。本课题的研究是一项具有重要理论意义和实际应用价值的工作，对现有的理论和算法有所发展。

Adaptive finite element method has already become one of the most effective methods for numerically solving scientific and engineering problems. Superconvergence and recovery techniques have the important application value and the prospect in a posteriori error estimation and adaptive finite element methods. This work is to promote the application of finite element superconvergence and recovery techniques in adaptive computations. We study the finite element superconvergence and value, gradient and flux recovery, construct the recovery type a posteriori error estimators based on different recovery techniques and design the adaptive algorithms. Compare with exist works, we shall relax the constraints of superconvergence on mesh, and propose a computable parameter to show the relationship between the mesh's quality and the convergence rate of error. We present an analytical investigation in the superconvergence properties on CVDT mesh, and design the mesh refine and optimization algorithm which can maintain the superconvergence. We also propose new recovery methods which are high accuracy, easy to implement and efficient. We design a new adaptive finite element method base on the new superconvergence results and new recovery techniques. This study is of great both theoretical and practical significance.