PDE solvers: Frequency-domain, time-domain and hybrids---with applications to materials science and engineering

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This effort concerns development of mathematical tools which can be used to make accurate predictions about physical phenomena, with impact on areas of significant societal interest such as electrical engineering (optics, electronics, photonics) communications (antennas), atmospheric science, medicine (tomography, imaging, diagnostic and therapeutic ultrasound, targeted drug delivery), military and civilian remote sensing (radar, sonar, stealth), renewable energy production and energy policy (Doppler signature of wind farms) etc. The methodologies to be pursued represent a change in paradigm in the mathematical approach. Oversimplifying for the sake of clarity, the new methods can be visualized as using some sort of a computational version of the french curve tool rather than a straight ruler in such a way that much more accurate representations of physical reality as well as greatly reduced computing costs result -- to the point that previously unfeasible simulations become possible. While such "smooth-curve representations" (or, in mathematical nomenclature, high-order/spectral methods) have been available for many years, the novelty of the proposed work is that it enables utilization of such highly accurate methodologies at vastly reduced computing costs and for highly complex engineering problems--such as those mentioned above--including complex electronic components, full vehicles, etc.Technically, this project concerns development and analysis of high-performance, highly accurate numerical algorithms for solution of Partial Differential Equations (PDE), with application to a wide range of problems in materials science and engineering. The proposed algorithms emphasize accuracy, efficiency as well as generally applicability on the basis of spectral and high-order methodologies; the associated theoretical discussions, in turn, seek to provide necessary background and performance guarantees. This effort considers two broad application areas, namely, I. Frequency-domain, time-harmonic acoustics and electromagnetism, and II. Time-domain PDE in general three-dimensional domains, with applicability to acoustics, electromagnetism and elasticity as well as compressible and incompressible fluid-dynamics.