Novel Approximation Techniques for Maxwell's Equations

这项研究的主要目的是发展和研究一种新的有限元负范数最小二乘法来求解电磁学中的麦克斯韦方程问题。计算电磁学是开发新技术的关键组成部分，其应用领域包括生物医学工程、微波工程、遥感和地下传感、天线分析和隐身技术。人们的注意力将集中在一种新的计算方法上，该方法基于各种问题的非常弱的公式。我们将研究这种新的计算方法在电磁场中的几个问题上的应用，包括涡流问题、时间谐波问题和所谓的时域问题。此外，还将从新的角度研究近场近似散射问题的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发挥着重要的作用。例如，设备应用涉及天线和波导、微电路和换能器的设计。电磁成像正被用于地球物理应用，以更好地了解和生产现有的油层，以及涉及肿瘤检测的医疗应用。本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稳健性。在医学应用中，更准确和可靠的计算可能会区分肿瘤和非肿瘤细胞，或者在雷达应用中，可能会让人区分敌人和朋友。

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.