偏微分方程最优控制问题的高精度低阶非协调有限元方法研究

Optimal control problems governed by partial differential equations have been widely used in ordinary living and engineering, such as atmospheric pollution control, temperature control, oil exploiting, image processing, and so on. Most of the optimal control problems always have large scale computations and need to be solved quickly, so it is important to study the corresponding high accuracy numerical methods. The nonconforming elements have more advantages than the conforming ones in the parallel computations. The exact solution of the optimal control problems always have low order regularities, so we should choose some low order elements to approximate the variables in the first place. The project aims to research the high accuracy low order nonconforming finite element methods and mixed finite element methods for optimal control problems. The main content includes the constructions of the elements, the error estimates, and the numerical experiments, etc. Firstly, by using the properties of the low order nonconforming elements, such as the consistency error being one order higher than the interpolation error, and the interpolation operator equaling to the Rieze projection, the optimal order error estimates and superclose results will be derived. Then, by employing the post-processing technique, the global superconvergence properties will be obtained. Lastly, some numerical experiments will be carried out to verify the theoretical analysis. The research provides some new choices for numerical computations of the optimal control problems, which have important theoretical significance and practical applications for expanding the applications of the nonconforming finite element methods.

偏微分方程最优控制问题在日常生活和工程领域中有着广泛应用，如大气污染控制、温度控制、石油生产和图像处理等方面。由于很多最优控制问题的计算规模十分巨大，对于求解速度的要求很高，所以研究其高精度数值算法就显得尤为重要。事实上，相对于协调元而言，非协调元在计算机并行计算中更具优势。而最优控制问题的解一般具有较低的正则性，因此应首选低阶元来逼近相关变量。本项目主要研究最优控制问题的高精度低阶非协调有限元和混合元逼近方法，其主要内容包括单元构造、误差估计及数值模拟等方面。首先利用低阶非协调元的一些特性，如相容误差比插值误差高一阶，插值算子与Rieze投影等价等，导出最优误差估计及超逼近结果，进而采用插值后处理技术，得到整体超收敛性质，最后给出一些数值算例来验证理论分析的正确性。该项目的研究为最优控制问题的数值计算提供了一种新的途径，对于拓宽非协调有限元方法的应用范围有着重要的理论研究意义和应用价值。

偏微分方程最优控制问题在日常生活和工程领域中有着广泛应用，比如大气污染控制、温度控制、石油生产和图像处理等方面。由于绝大多数最优控制问题的计算规模十分巨大，对于求解速度的要求很高，所以研究其高精度数值算法就显得尤为重要。本项目主要研究了最优控制问题的高精度低阶非协调有限元和混合元逼近方法，其主要内容包括单元构造、误差估计以及数值模拟等方面。首先利用了一些低阶非协调元的特殊性质，如相容误差比插值误差高一阶，插值算子与Riesz投影等价等，导出最优误差估计及超逼近结果，进而采用插值后处理技术，得到整体超收敛性质，最后给出一些数值算例来验证理论分析的正确性。该项目的研究为最优控制问题的数值计算提供了一种新的途径，这对于拓宽非协调有限元方法的应用范围有着重要的理论研究意义和应用价值。