目前自适应有限元方法的最优性和收敛性的理论，均是以自适应网格的有限元解为精确求解作为前提；然而，实际计算中对线性方程组的求解并不会采用精确求解的方式，因此，相应地，存在两个方面的问题：一方面是线性方程组的近似求解算法对整个自适应算法的最优性及收敛性的影响；另一个方面，如何快速地对自适应网格上对应的线性方程组进行求解。本项目将重点考察混合有限元方法。我们以二阶椭圆问题、Stokes问题、线弹性问题和界面问题为研究对象，旨在设计局部加密网格上混合有限元的快速求解算法，讨论对应的近似解的精度对整个自适应算法的影响，从而为混合有限元的自适应算法的实际应用提供有效工具和理论依据，完善自适应有限元算法过程。

The optimality and convergence theory for the adaptive finite element method is all established with the assumption that the finite element solution on each adaptive mesh is accurate. However, the exact solution of a linear equation is hard to obtain in actual calculation, so there exist two problems: on one hand, how the approximate solution of a linear equation affects the optimality and convergence of the entire adaptive algorithm; on the other hand, how to solve fast the linear equation based on the adaptive mesh. This project will focus on the mixed finite element method. Corresponding to second-order elliptic problem, Stokes problem, linear elasticity problem and interface problem, this project aims to design multigrid algorithm for mixed finite element method based on the local refined meshes, and to study the influence of the approximate finite element solution on the adaptive algorithm, thereby to provide an effective tool and theoretical basis for the practical application of adaptive mixed finite element method, and then to finish the adaptive finite element process.