Adaptive Discontinuous Galerkin Methods and Applications

科学计算，特别是计算数学被认为是科学进步的关键。通过取代昂贵的，有时甚至是不可能的实验室实验（如超新星爆炸），数值模拟作为有效的替代方法的重要性稳步增长。因此，最先进的算法和代码的发展，以及随之而来的对其预测的信心的增加，对所有这些领域的进展都很重要。该项目的研究项目旨在开发、分析和计算机实现数值方法，以近似一些控制应用数学、生物医学、工程和物理领域关键现象的偏微分方程的解。特别地，这项研究计划将被应用于生成有效的数值算法，通过关注其机电性质和由此产生的流体流动模式来有效地模拟心脏的功能。另一个对社会具有重要意义的应用将是将新开发的算法应用于地球气候的模拟，通过研究包括温室气体在内的各种化学物质以及气溶胶、灰尘和黑碳等污染物的影响。所获得的知识将通过研讨会和会议传播给其他研究人员。将开发新课程，学生将参与项目的各个阶段。后一个方面对于培养新一代研究人员至关重要。本文所描述的数值方法，即有限元法，特别是不连续伽辽金法，已经被研究了几十年。然而，在分析和应用领域都有足够的增长需求。将对传统的以及最近引入的后验误差估计进行研究，将这些估计扩展到这些方法无法达到的方程类别。这将导致改进和灵活的适应性方法，将用于解决具有实际重要性的问题。自适应方法的另一个应用将是回答有关非线性色散方程孤波解的存在性、唯一性和稳定性性质的解析性问题。求解由此产生的线性和非线性方程组的关键问题将构成所提出的研究的一个组成部分，多层次域分解和多网格方法构成了这一背景下推力的主干。考虑到所研究的大多数问题涉及三维空间和时间，在大规模并行计算机上的实现将被积极地追求。

Scientific computing and especially computational mathematics are recognized as crucial to the advancement of science. By replacing expensive and sometimes even impossible laboratory experiments (e.g., supernova explosions), numerical simulations have grown steadily in importance as effective alternatives to these. Therefore, the development of state of the art algorithms and codes, with the concomitant increase in confidence in their predictions, is important to progress in all these fields. The research program in this project aims at the development, analysis, and computer implementation of numerical methods designed to approximate the solutions of some partial differential equations that govern key phenomena in the fields of applied mathematics, biomedicine, engineering, and physics. In particular, this research program will be applied to generate efficient numerical algorithms for the effective simulation of the functioning of the heart by focusing on its electro-mechanical nature and the resulting fluid flow patterns. Another application of importance to society will be to apply the newly developed algorithms to the simulation of the Earth's climate by studying the influence of various chemicals including greenhouse gases, as well as pollutants such as aerosols, dust and black carbon. The knowledge gained will be communicated to other researchers through seminars and conferences. New courses will be developed and students will participate in all phases of the project. The latter aspect is crucial in developing the new generation of researchers.The numerical methodologies described in this proposal namely the finite element method and in particular the discontinuous Galerkin method have been studied for several decades now. Yet there is ample need for growth in both the analytical and application arenas. Research on traditional as well as recently introduced a posteriori error estimates will be conducted extending these estimates to classes of equations that had stayed beyond the reach of such methods. These will lead to improved and flexible adaptive methods that will be applied to solving problems of practical importance. An additional application of adaptive methods will be to answer questions of analytical nature concerning the existence, uniqueness, and stability properties of solitary wave solutions of nonlinear dispersive equations. The crucial problem of solving the resulting systems of linear and nonlinear equations will form an integral component of the proposed research with multilevel domain decomposition and multigrid methods forming the backbone of the thrust in this context. Implementation on massively parallel computers will be pursued actively given that most of the problems studied involve three spatial dimensions as well as time.