最优控制问题H1-Galerkin混合有限元方法研究

Optimal control problems have been widely met in all kinds of practical problems, such as, temperature control problems, electric field and magnetic field control problems, air pollution control problems, etc. Efficient numerical methods are among the keys to successful applications of optimal control in practical areas. Finite element approximation of optimal control problems plays a very important role in numerical methods for these problems. When the objective functional in the control problems contains the gradient of the state variables, mixed finite element methods should be used for discretization of the state equation. However, the technique of the classical mixed method leads to some saddle point problems whose numerical solutions have been quite difficult because of losing positive definite properties. In this project, we will investigate a priori error estimates, superconvergence and a posteriori error estimates for elliptic and parabolic optimal control problems by H1-Galerkin mixed finite element methods. Noting that this method has two notable advantages, one is that the method not only overcomes the inf-sup condition but the approximating finite element spaces are also allowed to be of different polynomial degree, the other is that velocity equation is separated from pressure equation and the discretized matrix is symmetric positive definite. Using this method, we can derive two approximations for the gradient of the primal state variable, one is the numerical approximation solution, the other is the derivative of the primal state approximation solution. Deriving the optimality conditions, error analysis and designing highly efficient algorithms are main difficulties.

最优控制问题存在于现实生活的各个方面，如温度控制问题、电磁场控制问题、空气污染控制问题等等。目前，已形成多种有效数值计算方法来求解这些问题，其中有限元方法应用最为广泛。当最优控制问题目标泛函包含标量函数的梯度时，混合有限元方法便是一种有效的数值方法。然而，采用标准的混合有限元方法进行求解时，离散系统容易产生鞍点问题。本项目将利用H1-Galerkin混合有限元方法来求解最优控制问题，主要研究椭圆和抛物最优控制问题的先验误差估计和超收敛以及抛物最优控制问题后验误差估计。此方法有两个优势，一是离散空间可以自由选取，不需要满足inf-sup条件，二是速度方程独立于压力方程，并且离散矩阵是对称正定的。关于速度变量，可以得到两个逼近解，一个是直接逼近解，另一个是对压力变量的数值解求导得到。最优性条件推导和误差分析以及高效算法设计是本项目的主要难点。

近年来，已形成多种有效数值计算方法来求解偏微分方程支配的最优控制问题，其中有限元方法应用最为广泛。当最优控制问题目标泛函包含标量函数的梯度时，混合有限元方法便是一种有效的数值方法。本项目主要利用H1-Galerkin混合有限元方法和RT混合有限元方法来求解椭圆和抛物最优控制问题，研究先验和后验误差估计。目前，在本项目的支持下，我们已经获得三项结果：. 1 给出椭圆最优控制问题H1-Galerkin混合有限元方法的最优性条件，获得所有变量的先验和后验误差估计。. 2 利用RT混合有限元方法来求解广义椭圆最优控制问题，主要分析所有变量的L^{2}和H^{-1}范数先验误差估计，并给出数值算例验证理论结果。. 3 给出抛物最优控制问题H1-Galerkin混合有限元方法的最优性条件，获得所有变量的先验和后验误差估计。. 前两项成果发表在国际SCI杂志：. 1 T. Hou. A priori and a posteriori error estimates of H^{1}-Galerkin mixed finite element methods for elliptic optimal control problems. Computers & Mathematics with Applications, 2015, 70 (10), 2542-2554.. 2 T. Hou and L. Li. Error estimates of mixed methods for optimal control problems governed by general elliptic equations. Advances in Applied Mathematics and Mechanics, 2016, 8 (6), 1050-1071.. 第二篇论文得到了审稿人较好的评价：. ''This is a good piece of work, and I would like to recommend it be published in the journal.''