Adaptive Discontinuous Galerkin Methods and Applications

在其广泛的轮廓中，该提案中概述的研究计划旨在开发，分析和计算机实现旨在近似某些偏微分方程的解的数值方法，这些偏微分方程在工程，物理，生物医学和数学领域具有重要应用。虽然不连续Galerkin方法将构成这项工作的核心方法，但将涵盖广泛的主题：先验和后验误差估计，自适应方法，使用多重网格和区域分解方法的线性和非线性系统的有效解决方案，灵活的数据结构和多核并行处理器上的实现，最后，应用于实际和分析问题。一个特别有吸引力的组成部分是通用的后验误差估计的发展。这些可以找到广泛的接受科学家谁不具有数学背景，是必要的建设传统的后验估计。一方面，算法和代码开发与另一方面，对其属性的严格分析之间将取得明智的平衡。 自近代以来，科学家们已经成功地通过使用偏微分方程（PDE）作为模型来描述自然现象。描述最大（宇宙）尺度现象的爱因斯坦方程，描述最小（原子）尺度现象的薛定谔方程和用于模拟大量流体流动的Navier-Stokes方程只是这些PDE中的一小部分，没有它们，现代世界就不会是今天的样子。然而，这样的偏微分方程几乎不可能精确求解，从一开始，科学家们就采用数值计算来近似未知的解。当然，在数值techinques的进步必须跟上PDE的进步，最近自适应数值技术已经成为一个非常有前途的工具，在处理即使是最困难的近似任务。在执行本提案中概述的研究项目时，P.I.将开发新的和有前途的自适应数值算法，分析其属性，并实现它们的最先进的计算机。这些方法和计算机代码将成为工具库的一部分，使工程师，物理学家和数学家能够理解他们所使用的特定PDE所描述的现象。例如，本文提出的一个研究活动是数值研究恶性肿瘤改变形状的方式。最后，为了保持培养下一代教师和研究人员的基本传统，一名研究生将参加这些项目，部分完成他的博士学位。℃下

In its broad outlines, the research program outlined in this proposal aims at the development, analysis and computer implementation of numerical methods designed to approximate the solutions of some partial differential equations with important applications in the fields of Engineering, Physics, the Biomedical Sciences and Mathematics. While the discontinuous Galerkin method will constitute the core methodology of this effort, a wide range of themes will be covered: A priori and a posteriori error estimates, adaptive methods, efficient solution of linear and nonlinear systems using multigrid and domain decomposition methods, flexible data structures and implementation on multicore and parallel processors and finally, applications to practical and analytical problems. A particularly appealing component is the development of generic a posteriori error estimators. These could find wide acceptability among scientists who do not have the mathematical background that is necessary to the construction of the traditional a posteriori estimators. A judicious balance will be struck between algorithm and code development, on one hand, and the rigorous analysis of their properties, on the other. Since the dawn of the modern era, scientists have been successful in describing natural phenomena by using partial differential equations (PDE's) as models. The Einstein equations that describe phenomena at the largest (cosmic) scales, the Schrodinger equation that describes phenomena at the smallest (atomic) scales and the Navier-Stokes equations that are used to model a plethora of fluid flows are but a few of such PDE's without which the modern world would not be what it is today. Yet such PDE's are almost impossible to solve exactly and since the beginning, scientists have resorted to numerical calculations to approximate the unknown solutions. Naturally, advances in numerical techinques must keep pace with advances in PDE's and quite recently adaptive numerical techniques have emerged as a very promising tool in tackling even the most difficult approximation tasks. In carrying out the research projects outlined in this proposal, the P.I. will develop novel and promising adaptive numerical algorithms, analyze their properties and implement them on state of the art computers. These methods and computer codes will become part of the arsenal of tools enabling engineers, physicists and mathematicians to understand the phenomena described by the particular PDE's they are using. For example, one research activity proposed herein is to numerically investigate the ways in which a malignant tumor changes shape. Finally, in maintaining the essential tradition of training the next generation of teachers and researchers, a graduate student will participate in these projects in partial fulfillment of his Ph.D. degree.