Weak Galerkin Modeling of Wave Scattering and Propagation in Dispersive Media

电磁波在色散和非均匀介质中的传播和散射在微波、太赫兹（太赫兹）和光波技术中具有重要意义。电磁器件的设计必须考虑色散介质的影响，如生物组织、岩石、土壤、雪、冰、等离子体、光纤、雷达吸收材料等。对这些效应的理解导致了具有独特特性的新设备。由于解的几何复杂性、界面不连续性、解的奇异性和时变性质，发展高效、准确的数值方法来求解色散介质问题仍然是计算数学中的一个具有挑战性的问题。本提案的目标是发展一种新的高效和精确的数值方法来模拟、分析和模拟色散和非均匀介质中的电磁波问题。在本项目中，研究者将基于弱伽辽金有限元法构建稳定可靠的公式，用于处理复杂几何、界面不连续和低正则解，然后将分析扩展到时间相关问题。基于介质色散模型和介电界面场分量的复杂耦合，将发展出新的数学格式。提出的方法将应用于一些现实问题，包括微波乳房成像、探地雷达和双负超材料。本项目开发的先进计算工具，包括构建有效的数值公式、处理复杂几何和界面不连续的计算工具，将在波在色散介质中的散射和传播的数学建模和计算方面取得重大进展。

The propagation and scattering of electromagnetic waves in dispersive and inhomogeneous media are of essential importance in microwave, THz (tera-hertz) and light-wave technologies. The effects of dispersive dielectrics, such as biological tissues, rocks, soil, snow, ice, plasma, optical fibers, and radar-absorbing materials, must be taken into account in the design of electromagnetic devices. The understanding of these effects leads to new devices with unique characteristics. Due to geometric complexity, interface discontinuities, solution singularity and time-varying nature of the solution, the development of efficient and accurate numerical methods for solving dispersive media problems remains a challenging issue in computational mathematics. The goal of this proposal is to develop a new efficient and accurate numerical method for modeling, analysis, and simulation of electromagnetic wave problems in dispersive and inhomogeneous media. In this project, the investigator will construct stable and reliable formulations based on the weak-Galerkin finite element method for handling the complex geometry, interface discontinuity and low regularity solution, and then extend the analysis to time dependent problems. New mathematical schemes will be developed based on dispersive media models and complicated coupling of field components at the dielectric interface. The proposed approaches will be applied to a number of realistic problems, including microwave breast imaging, ground penetrating radar and double negative metamaterial. Advanced computational tools developed in this project, including those for constructing efficient numerical formulation, handling complex geometries and interface discontinuities will lead to a major advance in mathematical modeling and computation of wave scattering and propagation in dispersive media.