Theory and Application of Numerical Methods for Partial Differential Equations

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9706949 Jinchao Xu The proposed research is on the development, analysis and applications of advanced numerical methods for solving partial differential equations arising in sciences and engineering. The focus of research is on efficient multigrid and domain decomposition methods and other relevant algorithms that are suited for parallel and high performance computers. One major object is to try to make multigrid and domain decomposition methods be more practical and more easily used. In particular, efficient and practical multigrid methods will be developed for unstructured grids and also for grids that are currently available from existing (commercial) finite element software. Special techniques under development include the agglomeration method and auxiliary space (grid) methods. Theoretical analysis will also be carried out to justify the efficiency of various methods and also to motivate the development of more sophisticated methods. A major portion of the proposed research will be devoted to the development of efficient multigrid methods for convection-dominated convection-diffusion problems, Navier-Stokes equations and hyperbolic equations and also to the theoretical justifications of the efficiency of some special multigrid methods which use some carefully designed domain decomposition methods as smoothers. Some of the proposed research will be carried out in collaboration with computational scientists and engineers to develop practical and efficient methods for solving some real life problems. Because of the advent of high performance computers, it becomes more and more feasible to use computers to simulate real life problems. In fact it has become a more and more common practice to use computer simulations to replace the traditional and oftentimes very expensive laboratory experiments. Most of the real life problems can be modeled and described by the so-called partial differential equations. These equations can become more and more complicated if we want to have more and more accurate and more realistic modeling of the real life situations. Hence it is a constant challenge to develop efficient numerical methods for solving these equations. The proposed research is precisely for the study of a class of most advanced numerical algorithms for effectively solving partial differential equations on parallel and high performance computers. Our research has been directly tired to various practical applications such as environment protection and the design of medical material and devise. In fact, for example, one numerical package that we have helped develop (based on research related to this proposal) have been adopted by U.S. EPA for environmental assessment and protection.