对流扩散最优控制问题的有限元算法研究

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 algorithms and adaptive algorithms, and to perform some numerical examples to illustrate the correctness of the theoretical analysis and the effectiveness of the numerical algorithms. The research of this project will help people to simulate engineering problems such as air pollution and waste water treatment in a more accurate and scientific manner and to understand the mechanism and morphology of these engineering problems. It has great significance to resolve these engineering problems.

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。

由对流扩散方程刻画的最优控制模型在空气污染、污水处理等工程问题中有着重要的应用，本项目的研究工作为这些工程问题的解决提供了科学理论依据，具有重要的理论价值和应用前景。. 自项目实施以来，我们根据项目工作计划，积极查阅文献资料，借鉴国内外相关研究成果，围绕几类最优控制模型开展了深入研究工作。首先，针对状态变量具有边界层或内部层，解的正则性低，建立了求解对流扩散最优控制问题的连续内罚有限元法、间断有限元法离散格式，运用凸分析、对偶论证、椭圆重构等技术分析了控制变量、状态变量的残量型后验误差估计，基于投影梯度等优化算法设计了自适应算法，通过算例验证了算法的有效性。其次，对状态受限的对流扩散最优控制问题采用Morea-Yosida和Larentiev正则化方法，建立了求解正则化问题的边界稳定化有限元离散格式，运用凸分析、对偶论证等技术分析了状态变量、控制变量的残量型后验误差估计。最后，研究了一类反常扩散最优控制问题和一类抛物方程狄利克雷边界控制问题的有限元逼近，建立了有限元离散格式，建立了控制变量和状态变量的先验误差估计，通过数值算例验证了理论分析的正确性。. 项目执行期间，项目组在Computers and Mathematics with Applications、International Journal of Computer Mathematics、Journal of Scientific Computing等国内外重要刊物上发表论文8篇，培养研究生4人，获得山东省自然科学基金省属高校优秀青年人才联合基金1项。