Analysis and Applications of the Discontinuous Galerkin Method

点击翻译按钮获取中文摘要

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.