The Development and Anlysis of Sweeping Preconditioners for Scattering Problems
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The main objective of this research is the development of efficient iterative methods for computing numerical approximations to the solutions of acoustic and electromagnetic scattering problems. A fundamental issue in any finite element approximation to a scattering problem is the efficient implementation of the far field boundary condition. In this research, this condition will be approximated using either finite elements with a ``Perfectly Matched Layer'''' (PML) far field approximation or finite element boundary integral formulations. The research will develop innovative new preconditioning techniques for the solution of the algebraic systems resulting from the above-mentioned approximation techniques. The preconditioners are constructed using H-matrix approximation combined with a sweeping algorithm. Electromagnetic scattering problems play an important role in the development of new technologies with applications in, for example, antenna and microwave engineering, remote sensing, stealth vehicle design, electromagnetic and plasmonic cloaking, biomedical engineering, and electromagnetic and seismic subsurface imaging, and are especially indispensable in locating and monitoring gas and oil reservoirs. Simulations capable of efficiently solving acoustic and electromagnetic scattering problems involve complicated geometry, often, with fine scale inhomegenities. It is the goal of this research to significantly reduce the computational costs associated with solving above problems, thus enabling more realistic and robust computer design technology for systems involving electromagnetic scattering.