Theory and Application of Numerical Methods for Partial Differential Equations

小行星9706949 拟议的研究是发展，分析和应用先进的数值方法来解决偏微分方程在科学和工程中出现。 研究的重点是有效的多重网格和区域分解方法和其他相关算法，适用于并行和高性能计算机。 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，更容易使用。 特别是，将开发有效和实用的多重网格方法，用于非结构化网格和现有（商业）有限元软件中的网格。 正在开发的特殊技术包括凝聚法和辅助空间（网格）法。 还将进行理论分析，以证明各种方法的效率，并促进更复杂方法的发展。 拟议的研究的一个主要部分将致力于发展高效的多重网格方法的对流为主的对流扩散问题，Navier-Stokes方程和双曲型方程，也对一些特殊的多重网格方法，使用一些精心设计的区域分解方法作为平滑的效率的理论依据。 一些拟议中的研究将与计算科学家和工程师合作进行，以开发解决一些真实的生活问题的实用和有效的方法。 由于高性能计算机的出现，用计算机来模拟真实的生活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、通常非常昂贵的实验室实验已经成为一种越来越普遍的做法。 大多数真实的生活问题都可以用所谓的偏微分方程来建模和描述。 如果我们想对真实的生活情况进行越来越精确和更逼真的建模，这些方程会变得越来越复杂。 因此，它是一个不断的挑战，以发展有效的数值方法来解决这些方程。 所提出的研究正是为了研究一类最先进的数值算法，有效地解决偏微分方程的并行和高性能计算机。 本课题的研究成果已直接应用于环境保护、医用材料与器件设计等实际应用中。 事实上，例如，我们帮助开发的一个数值包（基于与该提案相关的研究）已被美国EPA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.