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（例如，超新星爆炸），数值模拟的重要性稳步增长，作为有效的替代方案。因此，最先进的算法和代码的发展，伴随着他们的预测信心的增加，在所有这些领域的进展是重要的。该项目的研究计划旨在开发，分析和计算机实现数值方法，旨在近似解决一些偏微分方程的问题，这些偏微分方程控制应用数学，生物医学，工程和物理领域的关键现象。特别是，该研究计划将被应用于生成有效的数值算法，通过专注于其机电性质和由此产生的流体流动模式，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通过研究包括温室气体在内的各种化学品以及气溶胶、灰尘和炭黑等污染物的影响，将新开发的算法应用于地球气候的模拟。所获得的知识将通过研讨会和会议传达给其他研究人员。将开发新的课程，学生将参加项目的所有阶段。后一个方面是至关重要的，在发展新一代的researchers.The数值方法中所描述的建议，即有限元法，特别是间断Galerkin方法已经研究了几十年了。然而，在分析和应用领域都有足够的增长需求。研究传统的，以及最近推出的后验误差估计将进行扩展这些估计类的方程，已经超出了这种方法。这些将导致改进和灵活的适应性方法，将应用于解决具有实际意义的问题。自适应方法的另一个应用是回答关于非线性色散方程孤立波解的存在性、唯一性和稳定性的分析性质的问题。解决由此产生的线性和非线性方程组的系统的关键问题将形成一个整体的组成部分，提出的研究与多级区域分解和多重网格方法形成骨干的推力在这种情况下。在大规模并行计算机上的实施将积极进行，因为大多数研究的问题涉及三个空间维度以及时间。

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.