Theory and Application of Numerical Methods for Partial Differential Equations

9706949徐金超拟研究科学与工程中出现的偏微分方程解的高级数值方法的发展、分析与应用。研究的重点是高效的多重网格和区域分解方法以及其他适合于并行和高性能计算机的相关算法。一个主要的目标是试图使多重网格和区域分解方法更实用和更容易使用。特别是，将为非结构网格和现有(商业)有限元软件提供的网格开发高效和实用的多重网格方法。正在开发的特殊技术包括聚集法和辅助空间(网格)方法。还将进行理论分析，以证明各种方法的有效性，并鼓励开发更复杂的方法。这项研究的主要内容将是发展高效的多重网格方法来求解对流占优的对流扩散问题、Navier-Stokes方程和双曲型方程，并从理论上证明某些特殊的多重网格方法的有效性，这些方法使用一些精心设计的区域分解方法作为光滑器。一些拟议的研究将与计算科学家和工程师合作开展，以开发实用而有效的方法来解决一些现实生活中的问题。由于高性能计算机的出现，用计算机模拟现实生活中的问题变得越来越可行。事实上，使用计算机模拟来取代传统的、往往非常昂贵的实验室实验已经成为一种越来越普遍的做法。大多数现实生活中的问题都可以用所谓的偏微分方程组来建模和描述。如果我们想要对现实生活中的情况进行越来越准确和逼真的建模，这些方程就会变得越来越复杂。因此，发展有效的数值方法来求解这些方程一直是一个挑战。本文的研究正是为了研究一类在并行高性能计算机上有效求解偏微分方程组的最先进的数值算法。我们的研究已经直接应用于各种实际应用，如环境保护和医疗材料和器械的设计。事实上，例如，我们帮助开发的一个数字包(基于与该提案相关的研究)已被美国环保局采用，用于环境评估和保护。

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.