Topics in anisotropic mesh adaptation and application to anisotropic diffusion problems

研究人员将研究各向异性网格适应用于偏微分方程的数值解。这些研究将基于所谓的M-均匀网格方法的各向异性网格自适应，其中任何不均匀的网格生成为一个均匀的张量指定的度量。成功的做法，促进更好地了解现有的算法，并在开发新的方法。一个具体的应用领域是各向异性扩散问题的数值解。新的网格生成中使用的网格条件和度量张量已开发的方法，使有限元逼近一类各向异性扩散问题满足离散最大值原理，并表现出没有虚假的振荡和文物。研究课题包括将现有理论扩展到三维问题和时间相关问题，以及发展后验各向异性误差估计和度量张量。该项目涉及发展有效和可靠的方法，用于数值求解各向异性扩散问题和其他表现出各向异性特征的问题。这些问题产生于几个应用领域，包括等离子体物理学（聚变实验和天体物理学），地下水污染建模，石油储层模拟，图像处理和全球气候模拟，这些对我们国家的经济，环境和安全至关重要。不幸的是，标准的数值方法往往产生虚假的振荡和文物，并引入过多的数值耗散的数值解，由于在这些问题中的扩散的高度各向异性和异质性。在这个项目中，各向异性网格自适应将被用来克服这些困难。它是一种网格自适应类型，允许在整个物理域中更改网格元素的大小、形状和方向。研究人员在他的工作中证明，线性有限元解可以满足最大值原理，因此，当使用适当选择的各向异性网格时，不包含虚假振荡。本项目将沿着这条路线对各向异性网格自适应进行深入研究。 最大原理保持计划将开发时间相关的问题和三维问题，并将研究其应用到各向异性扩散问题。这些研究将有助于更好地理解各向异性网格自适应，并为应用问题的数值模拟提供有用的工具，包括具有许多重要应用的各向异性扩散问题。

The investigator will study anisotropic mesh adaptation for use in the numerical solution of partialdifferential equations. The studies will be based on a so-called M-uniform mesh approach of anisotropic mesh adaptation where any nonuniform mesh is generated as a uniform one in the metric specified by a tensor. Success has been made with the approach in facilitating a better understanding of existing algorithms and in developing new methods. A specific application area is the numerical solution of anisotropic diffusion problems. New mesh conditions and metric tensors for use in mesh generation have been developed with the approach so that finite element approximations to a class of anisotropic diffusion problems satisfy a discrete maximum principle and exhibit no spurious oscillations and artifacts. The research topics will include the extension of the existing theory to three dimensional problems and time dependent problems and the development of a posteriori anisotropic error estimates and metric tensors.The project is concerned with the development of efficient and reliable methods for the numerical solution of anisotropic diffusion problems and other problems exhibiting anisotropic features. Those problems arise from several application areas including plasma physics (fusion experiments and astrophysics), groundwater contamination modeling, petroleum reservoir simulation, image processing, and global climate simulation that are crucial to our national's economy, environment, and security. Unfortunately, standard numerical methods often produce spurious oscillations and artifacts and introduce excessive numerical dissipation in the numerical solution due to the highly anisotropy and heterogeneous nature of diffusion in those problems. In this project, anisotropic mesh adaptation will be employed to overcome these difficulties. It is a type of mesh adaptation that allows the size, shape, and orientation of mesh elements to change throughout the physical domain. The investigator has demonstrated in his work that a linear finite element solution can be made to satisfy the maximum principle and thus contains no spurious oscillations when a properly chosen anisotropic mesh is used. In-depth studies of anisotropic mesh adaptation will be carried out along this line in the project. Maximum-principle preserving schemes will be developed for time dependent problems and three dimensional problems and their application to anisotropic diffusion problems will be investigated. The studies will lead to a better understanding of anisotropic mesh adaptation and provide a useful tool for the numerical simulation of application problems, including anisotropic diffusion problems which have many important applications.