Novel Approximation Techniques for Maxwell's Equations

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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.