Analysis and Applications of the Discontinuous Galerkin Method

间断伽辽金法的分析与应用

• 批准号: 0411448
• 负责人: Ohannes Karakashian
• 金额: $ 11.52万
• 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: 2004
• 资助国家: 美国
• 起止时间: 2004-09-01 至 2008-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0411448&HistoricalAwards=false
• 关键词: Analysis; Applications; Discontinuous; Galerkin; Method

In its broader outlines, this research program aims at the development,analysis and computer implementation of numerical methods designed toapproximate the solutions of some partial differential equations that haveimportant applications in the fields of engineering and physics. Indeed,elliptic equations, the Navier-Stokes equations and nonlinear waveequations despite having been cultivated for decades, still offer fertileground for further exploration, for there are still a plethora ofunanswered questions and a pressing need for more efficient and fasteralgorithms. The discontinuous Galerkin method will constitute the core methodology ofthis effort. While going back to 1973, major interest did not focus on ituntil the nineties. Today it constitutes one of the most active areaswithin finite elements if not the whole range of methods for the numericaltreatment of partial differential equations. It has not been asextensively explored as the standard Galerkin version, yet what is knownso far offers a tantalizing glimpse of its potential. The project willinvolve various areas at the cutting edge of numerical analysis andscientific computing, in particular, the development of convergent andefficient adaptive methods designed to reduce the run time of thealgorithms by finding optimal or quasi-optimal meshes. These adaptivemethods will require continuing the work on the development of sharpa-posteriori error estimators designed to identify regions where thesolution is varying rapidly. Major efforts will be directed towardspursuing a recently identified strategy for reducing the number ofiterations that current adaptive algorithms require in achieving aprescribed level of accuracy.Scientific computing is recognized as crucial to the advancement ofscience. As such, the development of state of the art algorithms and codesis important for the progress of technology. Specifically, improvedadaptive codes resulting from this project will have impact on a widerange of problems and applications involving fluid flow phenomena, theelucidation of extremely fast chemical reactions by femtosecond lasers,and numerical simulations of supernova explosions.

从更广泛的角度来看，该研究项目旨在开发、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的解。事实上，椭圆方程、纳维-斯托克斯方程和非线性波动方程尽管已经研究了几十年，但仍然为进一步探索提供了沃土，因为仍然有大量未解答的问题，迫切需要更高效、更快的算法。间断伽辽金方法将构成这项工作的核心方法论。追溯到 1973 年，直到 90 年代才引起人们的主要兴趣。今天，它即使不是偏微分方程数值处理的全部方法，也是有限元中最活跃的领域之一。它尚未像标准伽辽金版本那样得到广泛的探索，但迄今为止已知的信息提供了对其潜力的诱人一瞥。该项目将涉及数值分析和科学计算前沿的各个领域，特别是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。这些自适应方法将需要继续开发锐帕后验误差估计器，旨在识别解快速变化的区域。主要努力将致力于追求最近确定的策略，以减少当前自适应算法在达到规定的精度水平时所需的迭代次数。科学计算被认为对科学的进步至关重要。因此，最先进的算法和代码的开发对于技术的进步非常重要。具体来说，该项目产生的改进的自适应代码将对涉及流体流动现象、飞秒激光器极快化学反应的解释以及超新星爆炸的数值模拟等广泛的问题和应用产生影响。