Frontiers of finite element methods

有限元方法是许多工程装置和自然现象模拟中不可缺少的工具。本文提出的研究旨在向各个方向拓展有限元的前沿。这些活动可分为四类。首先是在以前的国家科学基金会对混合有限元方法的支持下所进行的活动的延续，这导致了有限元设计的范式转变。由此探索了设计具有奇异性质的离散化的新机会。第二组活动针对高阶有限元的基本理论和实际问题。第三类涉及轴对称问题的数学上合理的有限元分析和构造。在第四类中，考虑了电磁学中的一些现代应用，这些应用可以受益于将其他技术与FEM相结合。这些活动的广泛影响包括许多可能受益的特定工程应用实例，包括纳米光子材料的模拟、天线的设计、心脏消融设备的工程，以及石油工业感兴趣的问题，如随钻测井和多孔介质流动。可以考虑如此广泛的应用，因为所提出的研究涉及广泛适用性的技术。例如，杂交技术的研究可以应用于表示各种物理现象的许多类型的方程。所提出的活动的另一个主题是设计有效的多层求解器，以解决涉及诸如散射和辐射等出射波的广泛实际问题。轴对称问题求解方法的研究影响着同轴电缆、波导、光纤等工程器件的仿真。高阶方法的研究活动不仅涉及应用数学，而且涉及纯数学分析。

Finite element methods (FEM) are an indispensable tool in the simulation of many engineering devices and natural phenomena. The research proposed here aims to expand the frontiers of FEM in various directions. The activities fall into four categories. The first is a continuation of the activities pursued under previous NSF support on hybridized finite element methods, which has resulted in a paradigm shift in FEM design. The consequent new opportunities in designing discretizations with exotic properties are explored. The second group of activities is tailored to fundamental theoretical and practical issues in high-order FEM. The third category deals with the analysis and construction of mathematically sound FEM for axisymmetric problems. In the fourth category, a number of modern applications in electromagnetics which can benefit from combining other techniques with FEM are considered.Broader impacts of the activities include many examples of specific engineering applications which can potentially benefit, including simulation of nanophotonic materials, design of antennas, engineering of devices for cardiac ablation, and problems of interest to the petroleum industry like logging-while-drilling and porous media flow. Such a range of applications can be considered because the proposed research involves techniques of broad applicability. For example, the research on hybridization techniques can be applied to many types of equations representing various physical phenomena. Another theme of the proposed activities is the design of efficient multilevel solvers for a wide range of practical problems involving outgoing waves such as scattering and radiation. The research on methods for axisymmetric problems can impact simulation of many engineering devices such as coaxial cables, wave guides, and optical fibers. The research activities on high-order methods touch upon questions that have implications not only in applied mathematics, but also in pure mathematical analysis.