Frontiers of finite element methods

有限元法（FEM）是模拟许多工程设备和自然现象的不可或缺的工具。 本文的研究旨在从多个方向拓展有限元的前沿领域。 这些活动分为四类。 第一个是在以前的NSF支持下进行的活动的延续杂交有限元方法，这导致了在FEM设计的范式转变。 随之而来的新的机会，在设计离散与异国情调的属性进行了探讨。 第二组活动是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可靠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，这些应用可以从其他技术与FEM的结合中受益。这些活动的更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广泛实际问题设计高效的多层求解器。 轴对称问题的求解方法的研究对同轴电缆、波导、光纤等工程器件的模拟具有重要的影响。 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.