对流扩散最优控制问题的有限元算法研究

项目名称： 对流扩散最优控制问题的有限元算法研究

项目编号： No.11301311

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 周兆杰

作者单位： 山东师范大学

项目金额： 23万元

中文摘要： 对流扩散最优控制模型在空气污染、污水处理等工程问题中有着广泛的应用，本项目旨在研究在上述问题中有重要应用背景的控制为逐点分布和状态受限的对流扩散最优控制问题。在这些问题中，状态方程的对流占优特性使得状态和伴随状态具有梯度变化剧烈的内部层或边界层，控制的逐点分布和状态受限则导致状态和伴随状态具有较低的正则性，这些性质给数值求解造成了极大的困难。关于这些问题的数值求解，一直为计算数学界与工程界所广泛关注。本项目将针对此类问题的特殊性，建立间断有限元和连续内罚有限元离散格式，运用凸分析、正则化、Green函数、对偶论证及Bubble函数等技术建立控制和状态的先验及后验误差估计，设计半光滑牛顿算法和自适应算法，对模型进行试算，验证理论分析的正确性和算法的有效性。本项目的研究有助于人们更科学、准确地模拟空气污染、污水处理等工程问题，深入了解这些问题的机理和形态，对这些问题的解决具有积极而重要的意义。

中文关键词： 对流扩散最优控制问题；逐点控制；状态受限；误差估计；数值算法

英文摘要： Convection diffusion optimal control models are widely used in some engineering problems such as air pollution problem and waste water treatment problem. The goal of this project is to investigate convection diffusion optimal control problems with pointwisely imposed control and state constraints. In these problems the state equation is usually convection dominated. This makes the state varible and adjoint state variable have interior layer or boundary layer with small widths, where their gradient changes rapidly. Pointwisely distributed control and state constraints cause the regularity of state variable and adjoint state variable lower than general distributed control problems. These properties bring great difficulty to numerical approximation of these kinds of convection diffusion optimal control problems. The numerical solution of such models has been widely concerned by the mathematical community and the engineering community. Aiming at these problems we plan to develop discontinuous Galerkin finite element and continuous interior penalty finite element discrete schemes, to derive a priori error estimates and a posteriori error estimates for state and control by using some techniques such as convex analysis, regularization, Green function, dual argument and bubble function, to design effective semi-Newton a

英文关键词： convection diffusion optimal control；pointwise control；state constraints；error estimate；numerical algorithm