The modeling of electromagnetic phenomena described by the Maxwell's equations is of critical importance in many practical applications. The numerical simulation of these equations is challenging and much more involved than initially believed. Consequently, many discretization techniques, most of them quite complicated, have been proposed. In this dissertation, we present and analyze a new methodology for approximation of the time-harmonic Maxwell's equations. It is an extension of the negative-norm least-squares finite element approach which has been applied successfully to a variety of other problems. The main advantages of our method are that it uses simple, piecewise polynomial, finite element spaces, while giving quasi-optimal approximation, even for solutions with low regularity (such as the ones found in practical applications). The numerical solution can be efficiently computed using standard and well-known tools, such as iterative methods and eigensolvers for symmetric and positive definite systems (e.g. PCG and LOBPCG) and reconditioners for second-order problems (e.g. Multigrid). Additionally, approximation of varying polynomial degrees is allowed and spurious eigenmodes are provably avoided. We consider the following problems related to the Maxwell's equations in the frequency domain: the magnetostatic problem, the electrostatic problem, the eigenvalue problem and the full time-harmonic system. For each of these problems, we present a natural (very) weak variational formulation assuming minimal regularity of the solution. In each case, we prove error estimates for the approximation with two different discrete least-squares methods. We also show how to deal with problems posed on domains that are multiply connected or have multiple boundary components. Besides the theoretical analysis of the methods, the dissertation provides various numerical results in two and three dimensions that illustrate and support the theory.

abstract: With the discovery of the Higgs Boson in 2012, particle physics has decidedly moved beyond the Standard Model into a new epoch. Though the Standard Model particle content is now completely accounted for, there remain many theoretical issues about the structure of the theory in need of resolution. Among these is the hierarchy problem: since the renormalized Higgs mass receives quadratic corrections from a higher cutoff scale, what keeps the Higgs boson light? Many possible solutions to this problem have been advanced, such as supersymmetry, Randall-Sundrum models, or sub-millimeter corrections to gravity. One such solution has been advanced by the Lee-Wick Standard Model. In this theory, higher-derivative operators are added to the Lagrangian for each Standard Model field, which result in propagators that possess two physical poles and fall off more rapidly in the ultraviolet regime. It can be shown by an auxiliary field transformation that the higher-derivative theory is identical to positing a second, manifestly renormalizable theory in which new fields with opposite-sign kinetic and mass terms are found. These so-called Lee-Wick fields have opposite-sign propagators, and famously cancel off the quadratic divergences that plague the renormalized Higgs mass. The states in the Hilbert space corresponding to Lee-Wick particles have negative norm, and implications for causality and unitarity are examined. This dissertation explores a variant of the theory called the N = 3 Lee-Wick Standard Model. The Lagrangian of this theory features a yet-higher derivative operator, which produces a propagator with three physical poles and possesses even better high-energy behavior than the minimal Lee-Wick theory. An analogous auxiliary field transformation takes this higher-derivative theory into a renormalizable theory with states of alternating positive, negative, and positive norm. The phenomenology of this theory is examined in detail, with particular emphasis on the collider signatures of Lee-Wick particles, electroweak precision constraints on the masses that the new particles can take on, and scenarios in early-universe cosmology in which Lee-Wick particles can play a significant role. / Dissertation/Thesis / Doctoral Dissertation Physics 2015