摘要
本文主要研究了两个问题,在文中分别加以阐述.下面就两部分内容作简单介绍.
文章的第一部分,探讨了稳态温控问题自适应有限元方法的收敛性.我们知道在实际应用中,自适应有限元方法(参见[2][5][6])常被用来数值求解最优控制问题.对本文而言,我们将致力于研究稳态温控问题.在文章的开端,首先讨论了当模型问题采用标准有限元逼近时,分别在L2-模和H1-模下得到的后验误差估计.然后,基于上述结果,我们得到了一个误差缩减率.更进一步,结合数据振荡的缩减率,可以构造模型问题的一个自适应有限元算法.在后续部分,我们证明了该算法是收敛的.
文章的第二部分,围绕气体在块状纳米材料中的压力扩散行为展开.由于任何固体材料都或多或少地能够渗透过一些气体,气体在固体中扩散的规律与气体自扩散公式有相同的形式.描述扩散行为的基本参量,即扩散系数是由气体一固体组合的性质决定.对于特定的气体,它在不同固体材料中的扩散系数是不同的.因此,对于未知固体材料,研究气体在其内部的扩散系数也能从一方面帮助人们了解它的结构、属性,这对工程领域的研究有着实际的意义.本文通过模拟气体在未知纳米材料块中的压力扩散实验,建立对应的偏微分方程模型,试图反演出气体在固体中的压力扩散系数.借助物理学相关知识,本文将反问题最终转化为最优化问题来处理.由于实验测量数据均为离散值,所以主体部分采用了偏微分方程数值解、数值积分(参见[29][30][31][32])的方法来处理.最优化问题拟用黄金分割搜索法(参见[26][28][33])求解.
In present paper, we focus on two problems. They will be discussed in two parts respectively.
In the first part, we concentrate on the Convergence of Adaptive Fi-nite Element Methods for Static Temperature Control Problem. Adaptive Finite Element Methods (see [2][5][6]) for numerically solving optimal control problems are often used in practice. In this part, we concentrate on the Static Temperature Control Problem. At the beginning, we discuss a posteriori error estimates for the standard finite element approxima-tion of the model problem in L2-norm and H1-norm respectively. Then based on the above results, we obtain an error reduction rate. Further-more, together with a reduction rate of data oscillation, we construct an adaptive FEM algorithm for the model problem. Next, we prove that this algorithm is convergent.
In the second part, we work on the Pressure diffusion behavior of the gas in nanomaterials. Gas can penetrate any solid matter more or less, and the diffusion rule of gas in solid is similar to its self-diffusion. Dif-fusion coefficient is the basic parameter, which is decided by the nature of the gas-solid combination. In different solid, the diffusion coefficient of gas is different as well. As an application, for an unknown kind of solid, to study the diffusion coefficient of gas in it can help us understand its structure. By simulating the pressure diffusion behavior of the gas in nanomaterials, we establish the corresponding partial differential equa-tions, and we try to calculate the diffusion coefficient. With knowledge of physics, the problem above can be transformed into optimization prob-lem. Because the data in this experiment is discrete, we have to solve the problem using numerical methods (see [29][30][31][32]). Optimization algorithm (see [26][28][33]) is needed also.
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