Analysis and Applications of the Discontinuous Galerkin Method

在其广泛的轮廓，研究计划的P.I.旨在发展、分析和计算机实现一些在工程和物理领域有重要应用的偏微分方程的数值方法。间断伽辽金方法是本文的核心方法论。本研究的最终目的是充分利用这一方法，通过寻找最优或准最优网格，发展收敛的、有效的自适应方法，以减少算法的运行时间.本文的另一个重点是发展区域分解和多重网格算法，以便在单机和多处理机上快速求解所得到的方程组。将开发方法和自适应算法的第二和第四阶椭圆问题，不可压缩的Navier-Stokes方程和theCahn-Hilliard方程，并使用它们来模拟现象由这些方程。自适应方法也将被用来模拟有限时间爆破的非线性发展方程。一些领域的应用是化学反应，其中域的几何形状起着重要的作用和肿瘤生长的模拟。科学计算作为“真实的生活”实验的一种经济有效的替代品，在科学的进步中发挥着越来越重要的作用，这些实验可能非常昂贵，比如飞机设计中的风洞实验，甚至不可能复制，比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。美国政府通过其各种资助机构，通过建立配备最新一代大规模并行计算机的超级计算中心进行了重要投资。为了充分利用这些机器的全部功能，其中一些机器拥有数万个独立的处理器，必须开发高效和“可扩展”的方法和算法，以跟上硬件的发展。事实上，大多数当前的算法都无法充分利用这些计算机的全部功能，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新方法，具有很大的应用潜力。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。进一步了解这种方法和发展的高效和并行的计算机代码将有积极的影响，不断增加的领域的科学，使numericalsimulation必不可少的使用.最后，两名研究生积极参与了这个项目，作为他们博士学位的部分实践。度要求。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.