Hybrid Methods for the Time Domain Integral Equations of Computational Electromagnetics

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The project "Hybrid methods for the time domain integral equations of electromagnetics" advances a multidisciplinary program consisting of basic numerical analysis of time domain integral equations (TDIEs) and a two-fold program of hybridization. The basic numerical analysis is devoted to issues of sparsification, quadrature error, error estimates and fast solvers for a new and exciting class of TDIE methods based on convolution quadrature (CQ-TDIE). The first prong of the hybridization work concerns combining CQ-TDIE with the volume finite element method. Not only would this hybridization allow for the easy simulation of inhomogeneous and complex media, but it also raises the tantalizing prospect of a perfect time domain integral based boundary condition. The second, and more speculative, hybridization approach combines CQ-TDIE with older Galerkin-based techniques with the aim of controlling dispersion and dissipation. It is anticipated that the resulting hybrid methods have the stability and spatial accuracy of CQ, but with the greater efficiency and reduced dispersion associated with Galerkin approaches. In particular, the technique would be ideally suited for the analysis of certain classes of modern technology involving long propagation distances through homogeneous regions (as occur in electromagnetic interference analysis) or mechanically moving parts (as necessary in many nanotechnology problems).Numerical simulation of physical processes reduces business prototyping costs, enables the safe prediction of the outcome of dangerous experiments, and can even make scientific discoveries by illuminating internal processes difficult to examine in the laboratory. In an era dominated by personal communication technology, the simulation of electromagnetic phenomena becomes especially crucial. "Hybrid methods for the time domain integral equations of electromagnetics" creates two new methods for the simulation of the interaction between electromagnetic fields and matter. While computerized approaches to electromagnetic analysis already exist, the methods to be created by the investigators allow for more accurate and efficient simulation of problems involving biological tissue, long propagation distances (such as occur in communications simulations), and mechanically moving parts (as needed in nanotechnology simulations). In addition to these benefits, the new method represents an important mathematical advance: it is based on a technique thought unstable for over three decades, which has only recently been made practical. The investigators therefore expect that the advances made in the analysis of electromagnetic problems under the auspices of this work will be easily translated to other scientific areas including acoustics, solid mechanics, and fluid mechanics.