Adaptive Discontinuous Galerkin Methods and Applications

本方案概述的研究计划旨在开发、分析和计算机实现数值方法，以近似求解一些在工程、物理、生物医学和数学领域具有重要应用的偏微分方程解。虽然不连续Galerkin方法将构成这项工作的核心方法论，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有效求解线性和非线性系统、灵活的数据结构和在多核和并行处理器上的实施，以及最后，在实际问题和分析问题上的应用。一个特别吸引人的部分是通用后验误差估计器的发展。这可能会在那些没有构建传统后验估计所必需的数学背景的科学家中找到广泛的接受性。一方面是算法和代码开发，另一方面是对它们的性质的严格分析，这将是一个明智的平衡。从现代开始，科学家就成功地用偏微分方程(PDE)作为模型来描述自然现象。描述最大(宇宙)尺度上的现象的爱因斯坦方程，描述最小(原子)尺度上的现象的薛定谔方程，以及用来对过多的流体流动进行建模的纳维尔-斯托克斯方程，都是这样的偏微分方程组中的一小部分，没有它们，现代世界就不会是今天的世界。然而，这样的偏微分方程组几乎不可能精确地求解，从一开始，科学家们就求助于数值计算来近似未知的解。当然，数值技术的进步必须与偏微分方程组的进步保持同步，最近自适应数值技术已经成为一种非常有前途的工具，即使在处理最困难的逼近任务时也是如此。在执行这项建议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，分析它们的性质，并在最先进的计算机上实现它们。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正在使用的特定偏微分方程所描述的现象。例如，这里提出的一项研究活动是对恶性肿瘤改变形状的方式进行数值研究。最后，为了保持培训下一代教师和研究人员的基本传统，研究生将参与这些项目，部分完成他的博士学位。

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.