Analysis and Applications of the Discontinuous Galerkin Method

概括地说，P.I.的研究计划旨在发展、分析和计算机实现一些在工程和物理领域具有重要应用的偏微分方程组(PDE)的近似解的数值方法。不连续伽辽金方法构成了这项工作的核心方法论。研究的最终目的是充分利用这种方法来开发收敛和高效的自适应方法，旨在通过寻找最优或准最优网格来减少算法的运行时间。另一项工作是开发区域分解和多重网格算法，以便在单处理机和多处理机计算机上快速求解所得到的方程组。P.I.将开发二阶和四阶椭圆型问题、不可压缩的Navier-Stokes方程和Cahn-Hilliard方程的方法和自适应算法，并用它们来模拟由这些方程建模的现象。自适应方法也将被用来模拟非线性发展方程的有限时间爆破。一些应用领域是化学反应，其中区域的几何形状扮演着重要的角色，以及对肿瘤生长的模拟。科学计算在科学的进步中发挥着越来越重要的作用，作为一种经济高效的替代方案，可以取代飞机设计中的风洞实验，甚至是无法复制的，如超新星爆炸和其他天体物理现象。值得一提的是，数值模拟在确定全球变暖的潜在影响方面发挥着至关重要的作用。美国政府通过其各种资助机构进行了重大投资，建立了配备最新一代大规模并行计算机的超级计算中心。为了充分利用这些机器的全部能力，其中一些机器拥有数万个单独的处理器，必须开发出高效和“可扩展”的方法和算法，以跟上硬件的进步。事实上，目前的大多数算法都不能充分利用这些计算机的全部能力，特别是当处理器的数量超过几千个时。间断Galerkin方法是近年来发展起来的一种具有很大潜力和广泛适用性的方法。它的许多特性包括灵活性、处理复杂几何图形的能力和可伸缩性。对这一方法的进一步理解以及高效和并行计算机代码的开发将对日益增长的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，以部分满足他们的博士学位要求。这将实现本项目的另一个目标，即为培养下一代研究人员做出贡献。

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.