Analysis and Applications of the Discontinuous Galerkin Method

间断伽辽金法的分析与应用

• 批准号: 0811314
• 负责人: Ohannes Karakashian
• 金额: $ 16.96万
• 依托单位: University of Tennessee Knoxville
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0811314&HistoricalAwards=false
• 关键词: Analysis; Applications; Discontinuous; Galerkin; Method

In its broad outlines, the research program of the P.I. aims at thedevelopment, analysis and computer implementation of numerical methods designedto approximate the solutions of some partial differential equations (pde's)that have important applications in the fields of engineering and physics. TheDiscontinuous Galerkin method constitutes the core methodology of thiseffort. The ultimate goal of the research is to make full use of this methodto develop convergent and efficient adaptive methods designed to reduce therun time of the algorithms by finding optimal or quasi-optimal meshes. Otherefforts will be directed towards the development of domain decomposition andmultigrid algorithm for the fast solution of the resulting systems of equationson single as well as multiprocessor computers.The P.I. will develop methods and adaptive algorithms for second and fourthorder elliptic problems, the incompressible Navier-Stokes equations and theCahn-Hilliard equations and use them to simulate phenomenamodeled by these equations. Adaptive methods will also be used to simulatefinite-time blowup of nonlinear evolution equations. Some areas ofapplications are chemical reactions where the geometry of the domain plays animportant role and the simulation of tumor growth.Scientific computing is playing an increasingly important role in the progressof Science as a cost effective alternative to "real life" experiments whichcould be very costly, say wind tunnel experiments in aircraft design, or evenimpossible to duplicate, such as supernova explosions and other astrophysicalphenomena. It is worth mention that numerical simulations are playing a crucialrole in the identification of the potential effects of global warming. The U.S.government, through its various funding agencies, has made important investmentsby the creation of supercomputing centers equipped with the latestgeneration of massively parallel computers. To extract the full power of thesemachines, with some having tens of thousands of individual processors, efficientand "scalable" methods and algorithms must be developed to keep pace with theadvances in hardware. Indeed, most current algorithms fall short of harnessingthe full power of these computers especially when the number of processorsexceeds a few thousand. The Discontinuous Galerkin method is a recentlyintroduced methodology with great potential and wide applicability. Its manyattributes include flexibility, ability to handle complex geometries andscalability. Further understanding of this approach and the development ofefficient and parallel computer codes will have a positive impact on theever increasing areas of Science that make essential use of numericalsimulations. Finally, two graduate students are actively participating in thisproject as partial fulfillment of their Ph.D. degree requirements. This willachieve another goal of this project which is to contribute to the training ofthe next generation of researchers.

总体而言，P.I. 的研究计划旨在开发、分析和计算机实现数值方法，这些方法旨在逼近一些在工程和物理领域具有重要应用的偏微分方程（pde）的解。间断伽辽金方法构成了这项工作的核心方法论。研究的最终目标是充分利用该方法来开发收敛且高效的自适应方法，旨在通过寻找最佳或准最佳网格来减少算法的运行时间。其他努力将致力于开发域分解和多重网格算法，以便在单处理器和多处理器计算机上快速求解所得方程组。将开发用于二阶和四阶椭圆问题、不可压缩纳维-斯托克斯方程和卡恩-希利亚德方程的方法和自适应算法，并使用它们来模拟由这些方程建模的现象。自适应方法还将用于模拟非线性演化方程的有限时间爆炸。一些应用领域是化学反应（其中域的几何形状起着重要作用）和肿瘤生长的模拟。科学计算作为“现实生活”实验的一种经济有效的替代方案，在科学进步中发挥着越来越重要的作用，“现实生活”实验可能非常昂贵，例如飞机设计中的风洞实验，甚至无法复制，例如超新星爆炸和其他天体物理现象。值得一提的是，数值模拟在识别全球变暖的潜在影响方面发挥着至关重要的作用。美国政府通过其各个资助机构进行了重要投资，建立了配备最新一代大规模并行计算机的超级计算中心。为了充分发挥这些机器的全部功能，其中一些机器拥有数万个单独的处理器，必须开发高效且“可扩展”的方法和算法，以跟上硬件进步的步伐。事实上，当前大多数算法都无法充分利用这些计算机的全部功能，尤其是当处理器数量超过数千个时。间断伽辽金法是最近提出的一种具有巨大潜力和广泛适用性的方法。它的许多属性包括灵活性、处理复杂几何形状的能力和可扩展性。对这种方法的进一步理解以及高效并行计算机代码的开发将对不断增加的必须使用数值模拟的科学领域产生积极影响。最后，两名研究生正在积极参与这个项目，作为他们博士学位的部分实现。学位要求。这将实现该项目的另一个目标，即为下一代研究人员的培训做出贡献。