A spectral finite element method on non-conforming meshes: Domain decomposition for high frequency scattering problems

Computational electromagnetics-the solution of Maxwell's equations using computers-is a key component of the modern design cycle for a wide variety of electrical engineering devices.These include, but are not limited to, antennas, microwave devices, photonic crystals, optical waveguides, and electric machines.This wide range of devices demonstrates the predictive power of the theory of electromagnetism and the need to accurately analyze Maxwell's equations in situations for which classical mathematical techniques are ineffective.This thesis describes a high accuracy finite element method suitable for solving Poisson and Helmholtz problems, which arise from Maxwell's equations.High accuracy finite element methods are particularly useful for high frequency electromagnetic scattering problems.This is because experimental and theoretical results regarding dispersion errors for finite element methods applied to the Helmholtz problem indicate that an effective approach to control dispersion is to increase the polynomial degree of the finite element model as a function of element size and frequency.Increasing the polynomial degree where solutions are smooth leads to high accuracy.However, there are difficulties associated with the solution of the resulting linear systems when the polynomial degree increases.This tends to limit the extent to which high degree polynomial modeling is adopted in practice.To circumvent these difficulties, this thesis develops, from first principles, a high accuracy one-dimensional finite element method that exploits Legendre polynomial expansions and the associated fast Legendre transform.The method is extended to higher dimensions, and implemented and tested in two dimensions, by developing a systematic approach to enforce inter-element continuity.This approach allows for both arbitrary refinement of local polynomial degree and non-conforming mesh refinement.The method proposed in this thesis is capable of computing solutions to a user specified tolerance-potentially as stringent as machine precision-efficiently.All element-wise computations are performed with near linear computational complexity, which allows for the use of high polynomial degree to achieve high accuracy.The developed method is efficient because it consists of a domain decomposition method that fully exploit these fast elementwise computations.As long as the coupling between domains in the decomposition increases in such a way as to control dispersion errors, the method can be applied to compute high accuracy solutions while only solving systems that are much smaller than the total number of unknowns.The thesis demonstrates this behavior on several electromagnetic problems, including beam steering by lenses and photonic crystal waveguides, and radar cross section computation for dielectric, perfect electric conductor, and electromagnetic cloak scatterers.

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