Topics in anisotropic mesh adaptation and application to anisotropic diffusion problems

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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.