李 若
论文题目：移动网格方法及其应用
作者简介：李 若，男，1973年07月出生，1996年09月师从于北京大学张平文教授，于2001年06月获博士学位。
摘 要
在广泛的实际应用问题中，往往出现解的性质相对恶劣，方程在求解区域的局部变化非常剧烈，或者是求解区域整体相对较大，却又要对其中小部分上解的细节信息要求很高的情况。对于这样的问题，在均匀的网格上求解是不现实的，尤其是高维的问题，计算量远远超出硬件的能力。自适应方法是解决这种问题的一个途径。移动网格方法，作为自适应方法的一种，主要是为了解决发展方程的计算问题而设计的方法，已经有二十多年的研究历史。本文主要是阐述了一种基于调和映射的移动网格方法。在移动网格方法中，需要引进一个逻辑区域作为参考，网格的移动往往通过一个区域变换来实现，网格可能发生缠绕的问题是移动网格方法中一个一直没有解决的问题，我们利用调和映射来构造区域间的变换，使得变换的存在唯一性有了理论的保证，这就为避免网格缠绕打下了基础。我们进而引进了一个迭代的过程来实现网格的移动，避免了数值原因导致的网格缠绕，彻底地解决了网格缠绕的问题。我们将网格移动和方程求解完全分开，从而使得移动网格方法在各类不同的问题中的应用，被完全归结为构造控制函数的问题，并且有利于程序开发。我们还设计了一个在不同的网格之间进行插值的格式，用来将解函数在不同的网格间过渡，数值结果表明这个格式和多项式插值方法相比，具有比较好的性质。我们用一系列比较难于处理的典型发展问题进行了计算，都取得了比较好的效果。计算结果说明，这样的方法产生的网格，变化的幅度可以很大，灵活性高，能够处理一些很复杂的情况，鲁棒性强。在网格移动以后，解的误差获得了有效的减少。我们发展的方法可以自然的在三维情况下实现，并且给出了数值算例。三维问题的移动网格方法，在以前是很少看见结果的。边界网格和内部网格偶合移动的问题，是一个实践中要求非常迫切，却又一直没有结果的问题。长期以来，只能对边界当成一维问题进行特殊处理。在二维的情况下，为了能够比较系统的处理边界节点的移动问题，我们扩大区域变换的求解空间，给出了一个将内部节点移动和边界节点移动偶合的格式，在一个扩大的空间里构造一个调和映射，将问题转换为一个边界控制问题，并且有存在唯一性的保证。在一定的情况下，这样的方法还能够在三维实现，并能推广到更一般的情况，将边界特殊处理的方法是不能做这样的推广的。在边界固定的情况下，构造网格移动的方向是比较容易实现的，但是在将边界和内部偶合的情况下，构造出网格移动的方向，我们的这个工作是有创新意义的。和边界固定的方法相比，使用这种偶合模式产生的网格，网格的质量更高，进一步的减小了解的误差，得到的网格的性态，显示出一种内蕴的合理性。我们还将移动网格方法应用于静态问题，并在其中研究使用后验误差估计来构造控制函数的方法。尽管加密的方法几乎总是根据后验误差估计来进行自适应，但对于移动网格方法来说，我们还没有看到使用后验误差估计构造控制函数的文献。我们得到的静态问题的数值结果，说明了用于构造控制函数的后验误差估计必须具有足够的精度才能够构造出能够获得满意的网格的控制函数，这些问题包括变分不等式问题和椭圆的最优控制问题，我们对这两个问题给出了能够用于构造控制函数的后验误差估计。
本文后面的内容组织如下：第零章介绍需要用到的数值计算方法的一些基本知识，第一章介绍几种自适应方法的关系和历史，第二章介绍基于调和映射的移动网格方法的基本格式，第三章介绍二维带边界的移动网格方法，第四章介绍变分不等式和最优控制问题的后验误差估计，并使用后验误差估计构造控制函数的技术，第五章是二维发展问题的算例，第六章是变分不等式和最优控制问题的算例，第七章是带边界二维问题的算例，第八章是三维问题的算例，第九章是总结和下一步工作的计划，附录A中是一个关于移动网格方法的误差估计的结果。
Abstract
A variety of physical and engineering problems develop dynamically singular or nearly singular solutions in fairly localized regions. In these problems, we are only interested in high resolutions in fairly small solution domain. With uniform meshes, the amount of computational time is too large to enable us to obtain useful numerical approximations, particular in multi-dimensions. Therefore, developing effective and robust moving mesh methods for these problems becomes necessary. Successful implementation of the adaptive strategy can increase the accuracy of the numerical approximations and also decrease the computational cost. In this thesis, we will investigate a class of moving mesh method which is based on harmonic mapping. A logical (or computational) domain is used as a reference and the mesh moving is implemented according to an appropriate domain transformation. With some traditional moving mesh methods, the meshes in physical domain may be tangled. To avoid this, the transformations are constructed based on harmonic mapping. A good feature of the adaptive methods based on harmonic mapping is that existence, uniqueness and non-singularity for the continuous map can be guaranteed from the theory of harmonic maps. Such theoretical guarantees are rare in the field of adaptive mesh generation.
In practice, there are three types of adaptive methods using finite element approach, namely the h-method, p-method, and r-method (i.e. moving mesh method). In the h-method, the overall method contains two parts, i.e. a solution algorithm and a mesh selection algorithm. These two parts are independent in the sense that the change of the underlying partial differential equations (PDEs) will affect the first part only. However, in some of the existing $r$-method, these two parts are strongly associated with each other and as a result any change of the PDEs will result in the rewriting of the whole code. In this work, we will propose a moving mesh method which also contains two parts, a solution algorithm and a mesh-redistribution algorithm. Our efforts are to keep the advantages for the r-method (e.g., keep the number of nodes unchanged) and for the h-method(e.g., the two parts in the code are independent). The mesh-moving is a procedure of iteration to construct the harmonic map between the physical mesh and the logical mesh. Each iteration step is to move the mesh closer to the harmonic map. A new scheme to interpolate the approximate solution from the old mesh into the new mesh is designed.
The numerical schemes are applied to a number of test problems in two- and three-dimensions. It is observed that the mesh-redistribution strategy based on the harmonic maps adapts the mesh extremely well to the solution without producing skew elements for the multi-dimension computations. There have been very few moving mesh results for three-dimensional problems. One of the main difficulties in dealing with the 3D problems is the treatment of the boundary grid re-distribution. In 2D, this difficulty can be handled by solving 1-D moving mesh equations on boundaries. However, the extension of this boundary grid redistribution technique to 3D is very difficult. In order to handle this problem, we will solve a constrained optimization problem that links the interior and boundary points as whole. It turns out that under certain conditions this scheme can be implemented in three dimensions. The mesh generated by the new scheme has higher quality than that generated by moving mesh methods with fixed boundaries.
We also applied the moving mesh method to some variational inequality and elliptical control problems. The key idea is to construct the monitor function by using appropriate a posterior error estimates. At the heart of any adaptive finite element refinement schemes are some appropriate a posteriori error estimators. The decision of whether further refinement of meshes is necessary is based on the estimate of the discretization error. If further refinement is to be performed then the a posteriori error estimators are used as a guide as to show the refinement might be accomplished most efficiently. Now the natural question is can we use them as monitor functions in the moving mesh methods? One of the objectives of this thesis is to address the above question. It is shown that the moving mesh methods with appropriate monitor functions can effectively solve elliptic obstacle problems and elliptic control problems.
The thesis is organized as follows. In Chapter 0, we list some basic results to be used in the thesis. Chapter 1 provides some introduction to adaptive finite element method. In Chapter 2, the moving mesh method based on harmonic mapping will be investigated. Chapter 3 is devoted to the discussions of the moving mesh method with boundary redistribution. In Chapter 4, we consider the application of the moving mesh methods for variational inequality and elliptic optimal control problems, with particular attention to the construction of the monitor functions. Chapter 5 presents some numerical computations for partial differential equations, while Chapter 6 shows the numerical computations for variational inequality and elliptic optimal control problems. In Chapter 7, we present numerical results for two-dimensional problems with boundary grid-redistribution. Numerical results for three-dimensional computations are presented in Chapter 8. Some concluding discussions are given in the final chapter. In appendix A, a symmetric error estimate for the moving mesh method will be briefly demonstrated.
