Novel Approximation Techniques for Maxwell's Equations

本研究的主要目的是开发和研究一种新的有限元负范数最小二乘方法，涉及电磁学中的麦克斯韦方程组的问题。计算电磁学是开发新技术的一个关键因素，其应用领域包括生物医学工程、微波工程、遥感和地下传感、天线分析和隐形技术。 注意力将集中在一个新的计算方法的基础上非常弱的配方的各种问题。 将对这种新的计算方法在电磁学中的几个问题，包括涡流问题，时间谐波问题和所谓的时域问题的应用进行调查。 此外，从新的角度研究了散射问题的近场近似的Maxwell-Berenger PML模型。 计算电磁学在器件设计、无损检测和成像等许多重要技术领域中发挥着重要作用。 器件应用涉及例如天线和波导、微电路和换能器的设计。电磁成像正被用于地球物理应用中，以更好地了解和生产现有的油藏，并在涉及肿瘤检测的医疗应用。该研究项目使用先进的分析和数值技术来提高大规模电磁计算的可靠性和鲁棒性。 在医学应用中，更精确和可靠的计算可能使肿瘤和非肿瘤细胞之间的区别成为可能，或者在雷达应用中，可能使人们能够区分朋友或敌人。

The main objective of this research is the development and study of a new finite element negative norm least-squares approach to problems involving Maxwell's equations in electromagnetics. Computational electromagnetics is a critical ingredient in development of new technologies with applications in, for example, biomedical engineering, microwave engineering, remote and subsurface sensing, antenna analysis and stealth technology. Attention will be focused on a new computational approach based on very weak formulations of various problems. Investigation of the application of this new computational approach to several problems in electromagnetics including eddy current problems, time-harmonic problems and the so-called time domain problem will be carried out. In addition, the Maxwell-Berenger PML model for the near-field approximation of the scattering problem will be investigated from the new point of view. A PML coordinate-stretching approach for treating exterior elliptic and hyperbolic problems will also be investigated.Computational electromagnetics plays an important role in many important technological areas including device design, nondestructive testing and imaging. Device applications involve, for example, the design of antennas and waveguides, microcircuits and transducers. Electromagnetic imaging is being employed in geophysical applications to better understand and produce existing oil reservoirs and in medical applications involving tumor detection. This research project uses advanced analytical and numerical techniques to improve the reliability and robustness of large scale electromagnetic computations. More accurate and reliable computations in medical applications might enable the differentiation between tumor and non-tumor cells or in radar applications might enable one to distinquish between friend or foe.