Frontiers of finite element methods

有限元方法是模拟许多工程设备和自然现象不可缺少的工具。本文提出的研究旨在多方向拓展有限元的研究领域。这些活动分为四类。第一个是在以前NSF支持下开展的杂交有限元方法的活动的继续，这导致了有限元设计的范式转变。探索了由此产生的具有奇异性质的离散化设计的新机会。第二组活动针对高阶有限元的基本理论和实践问题而量身定做。第三类是关于轴对称问题的数学合理有限元的分析和构造。在第四类中，考虑了一些现代电磁学应用，这些应用可以从其他技术与有限元的结合中受益。这些活动的广泛影响包括许多可能受益的特定工程应用的例子，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。可以考虑这样一系列的应用，因为拟议的研究涉及广泛适用的技术。例如，对杂交技术的研究可以应用于代表各种物理现象的许多类型的方程。拟议活动的另一个主题是为涉及散射和辐射等传出波的各种实际问题设计有效的多级解算器。轴对称问题的研究对同轴电缆、光波导及光纤等多种工程设备的模拟都有重要影响。高阶方法的研究活动涉及的问题不仅涉及应用数学，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.