受时变对流扩散方程约束的最优控制问题的SUPG方法研究

项目名称： 受时变对流扩散方程约束的最优控制问题的SUPG方法研究

项目编号： No.11201485

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 付红斐

作者单位： 中国石油大学（华东）

项目金额： 23万元

中文摘要： SUPG方法是近年来广受欢迎的用于求解对流扩散方程的数值手段之一。本项目通过合理选取稳定化参数，将SUPG方法应用到时变对流扩散最优控制问题中，并建立起一系列可靠、稳定的最优控制数值算法。具体研究内容如下：(1)通过适当选取依赖于时间步长的稳定化参数，建立向后欧拉SUPG最优控制全离散格式，推导出离散最优性条件。讨论算法的稳定性及其能量模先验误差估计，并通过数值模拟来验证理论分析的有效性。(2)在稳定化参数依赖于空间步长的情况下，建立半离散SUPG最优控制格式及其半离散最优性条件，进行L^2模及材料导数模意义下的收敛阶误差分析。(3)在稳定化参数依赖于空间步长情况下，对(1)中建立的SUPG最优控制全离散格式给出不同的先验误差估计，最后依然用数值实验进行验证。以上稳定化参数的选取及其SUPG最优控制数值算法的建立，将对时变对流扩散最优控制问题的数值方法研究提供新的思路和理论依据。

中文关键词： 最优控制问题；稳定化混合元方法；特征有限元方法；最小二乘方法；分裂方法

英文摘要： It is well known that SUPG method is currently one of the most popular numerical methods for solving convection-diffusion equations. By reasonable choose of stabilization parameters, this project aims to apply the SUPG method to optimal control problems subject to time-dependent convection-diffusion equations, and establish a series of reliable and stable numerical algorithms. In particular, the following aspects are discussed. (1) under the stabilization parameters depend on the length of the time step, we construct a fully discrete SUPG scheme combined with the backward Euler method in time, and deduce the discrete optimality conditions. Then stability bounds and a priori error estimates are derived based on energy arguments. Finally,numerical tests are given to support the theory.(2) under the stabilization parameters depend on the length of the space step, a semi-discrete SUPG scheme and the corresponding optimality conditions are derived. An error estimate for the L^2 norm and the norm of material derivative is obtained. (3) under the above choice of stabilization parameters and the constructed fully discrete SUPG scheme in part (1), another different error estimates are given. Finally, numerical examples validate the theoretical analysis. In all, this project provides a new idea to the study of numerical m

英文关键词： optimal control problems；stabilized mixed finite element method；characteristic finite element method；least-squares method；splliting method