Analysis and Applications of the Discontinuous Galerkin Method

从更广泛的范围来看，这项研究计划旨在开发、分析和计算机实现数值方法，旨在近似求解在工程和物理领域中有重要应用的一些偏微分方程解。事实上，椭圆型方程、Navier-Stokes方程和非线性波动方程尽管已经被培育了几十年，但仍然为进一步探索提供了肥沃的土壤，因为仍然有太多的问题没有答案，而且迫切需要更有效和更快的算法。不连续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，人们的主要兴趣才集中在它上面。今天，它构成了有限元中最活跃的领域之一，如果不是偏微分方程组的全部数值处理方法的话。它还没有像标准的Galerkin版本那样被广泛探索，但到目前为止所知的是，它的潜力提供了一个诱人的一瞥。该项目将涉及数值分析和科学计算的各个前沿领域，特别是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。这些适应性方法将需要继续开发夏普-后验误差估计器，以识别解决方案快速变化的区域。主要的努力将集中在最近确定的一项战略上，该战略旨在减少当前的自适应算法在实现所述精度水平时所需的迭代次数。科学计算被认为是科学进步的关键。因此，最先进的算法和编码的发展对技术的进步很重要。具体地说，由该项目产生的改进的自适应程序将对涉及流体流动现象、阐明飞秒激光极快化学反应以及超新星爆炸的数值模拟的更广泛的问题和应用产生影响。

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.