This project is concerned with the adaptive generation of anisotropic meshes for use in the numerical solution of partial differential equations. The goal is to study anisotropic meshes and their quality measures based upon rigorous mathematical analysis of the interpolation error and ultimately to develop efficient and reliable algorithms and computer software for their generation in two and three dimensions. At the core of mathematical modeling in science and engineering is a system of partial differential equations. Numerical simulation has become an indispensable tool for solving partial differential equations and therefore simulating physical processes. Meshes and their adaptive generation play a vital role in improving the accuracy and efficiency of numerical simulations. The key to mesh adaptation is the ability to simultaneously control the shape and the size of mesh elements. Research has so far concentrated on traditional or isotropic mesh adaptation where the element size is adapted to some error estimate while the shape is kept close to being equilateral. It is known that isotropic meshes tend to concentrate too many nodes in the regions of large solution error. This drawback can be overcome and the efficiency and accuracy of numerical simulations can be dramatically enhanced by properly choosing an anisotropic mesh where the elements are allowed to have a large aspect ratio but aligned with the solution. The research of this project will focus on the development of general anisotropic estimates for interpolation error and the study of their use in construction of new algorithms and computer codes for generating adaptive anisotropic meshes in two and three dimensions. Mesh assessment strategies and quality measures in geometry, alignment, and adaptation will be developed. The methods will be applied to the numerical simulation of groundwater flow systems where chemical contamination and other substances can cause the flow density to vary considerably. The successful completion of this project will greatly improve our understanding of the mathematics within mesh adaptation, and the outcome will serve as building blocks for further studies and development of mesh adaptation algorithms and software. The developed meshing strategy can be combined with many existing numerical methods and computer codes in other areas of science and engineering to greatly enhance their efficiency and accuracy. Education will be important part of this project. Students will be actively involved and receive their professional and interdisciplinary training during the term of the project.
