Analysis and Applications of the Discontinuous Galerkin Method

在其更广泛的轮廓，这个研究计划的目的是发展，分析和计算机实现的数值方法，旨在近似的解决方案，一些偏微分方程，在工程和物理领域的重要应用。实际上，椭圆方程、Navier-Stokes方程和非线性波动方程尽管已经被研究了几十年，但仍然为进一步的探索提供了肥沃的土壤，因为仍然有大量的问题没有得到解答，并且迫切需要更有效和更快的算法。间断Galerkin方法将构成这项工作的核心方法。而追溯到1973年，主要的兴趣并没有集中在它，直到九十年代。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数值处理方法的全部范围.它还没有像标准的伽辽金模型那样被广泛地探索，然而迄今为止所知的东西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前沿的各个领域，特别是收敛和高效自适应方法的发展，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工作的sharpa后验误差估计的发展，旨在确定地区的解决方案是迅速变化。主要的努力将集中在推动一个最近确定的策略，以减少当前自适应算法在达到预定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，最先进的算法和代码的发展对于技术的进步是重要的。具体而言，改进的自适应代码从这个项目产生的将有广泛的问题和应用程序的影响，涉及流体流动现象，飞秒激光的极快的化学反应的阐明，和超新星爆炸的数值模拟。

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.