Frontiers of finite element methods

University Of Florida, Gainesville FL

Investigators

Abstract

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.