Topics in anisotropic mesh adaptation and application to anisotropic diffusion problems

研究者将研究各向异性网格自适应在偏微分方程数值解中的应用。研究将基于所谓的各向异性网格自适应的m -均匀网格方法，其中任何不均匀网格在张量指定的度量中被生成为均匀网格。这种方法在促进对现有算法的更好理解和开发新方法方面取得了成功。一个特定的应用领域是各向异性扩散问题的数值解。利用该方法开发了用于网格生成的新网格条件和度量张量，使一类各向异性扩散问题的有限元近似满足离散极大值原理，并且没有虚假振荡和伪影。研究课题将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。该项目关注于发展有效和可靠的方法来数值求解各向异性扩散问题和其他具有各向异性特征的问题。这些问题来自等离子体物理（聚变实验和天体物理）、地下水污染模拟、油藏模拟、图像处理和全球气候模拟等几个应用领域，这些领域对我们国家的经济、环境和安全至关重要。不幸的是，由于这些问题的扩散具有高度的各向异性和非均质性，标准数值方法往往会产生虚假振荡和伪影，并在数值解中引入过多的数值耗散。本课题将采用各向异性网格自适应来克服这些困难。它是一种网格自适应，允许网格元素的大小、形状和方向在整个物理域内改变。研究者已经在他的工作中证明，当适当选择各向异性网格时，可以使线性有限元解满足最大原理，从而不包含杂散振荡。本项目将沿着这条路线开展各向异性网格自适应的深入研究。我们将为时间相关问题和三维问题开发最大原则保持格式，并研究它们在各向异性扩散问题中的应用。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.