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The propagation of electromagnetic fields is described by the vector Helmholtz equation. Finite element analysis of the vector Helmholtz equation involves complications such as spurious modes that do not arise in the scalar Helmholtz equation. Both driven vector field problems as well as vector eigenvalue problems may be corrupted by these divergent, nonphysical fields. Additionally, boundary and interface conditions in vector field problems are more complex than in scalar problems. In this thesis, the cause of spurious modes is analyzed and a condition called the inclusion condition is shown to eliminate the spurious modes. Mixed order covariant projection finite elements are shown to avoid spurious corruptions in both driven and eigenvalue problems. The proposed elements do not involve globally imposed constraints or penalty functions, and boundary and interface conditions are easily imposed.

Let K be a real quadratic number field. Let p be a prime which is inert in K. We denote the completion of K at the place p by Kp. Let ƒ > 1 be a positive integer coprime to p. In this thesis we give a p-adic construction of special elements u(r, ??) ∈ Kxp for special pairs (r, ??) ∈ (ℤ/ƒℤ)x x Hp where Hp = ℙ¹(ℂp) ℙ¹(ℚp) is the so called p-adic upper half plane. These pairs (r, ??) can be thought of as an analogue of classical Heegner points on modular curves. The special elements u(r, ??) are conjectured to be global p-units in the narrow ray class field of K of conductor ƒ. The construction of these elements that we propose is a generalization of a previous construction obtained in [DD06]. The method consists in doing p-adic integration of certain ℤ-valued measures on ??=ℤpxℤp pℤpxpℤp . The construction of those measures relies on the existence of a family of Eisenstein series (twisted by additive characters) of varying weight. Their moments are used to define those measures. We also construct p-adic zeta functions for which we prove an analogue of the so called Kronecker's limit formula. More precisely we relate the first derivative at s = 0 of a certain p-adic zeta function with -logₚ NKp/Qp u(r, ??). Finally we also provide some evidence both theoretical and numerical for the algebraicity of u(r, ??). Namely we relate a certain norm of our p-adic invariant with Gauss sums of the cyclotomic field Q (zetaf, zetap). The norm here is taken via a conjectural Shimura reciprocity law. We also have included some numerical examples at the end of section 18.

In this thesis, the problem of calculating power transmission line induced or radiated electromagnetic fíelds is studied by decomposing the line into a large number of snall segments, k¡own as Hertzian dipoles. Since the presence of lossy ground makes the electromagnetic fìeld distribution very different from that of the same dipole over perfectly conducting ground, different approaches should be followed for the lossy and lossless ground assumptions. The contribution of each dipole on the total electric and magnetic field is calculated using three analytical techniques. Two of these methods are frequency-domain solutions for the problem of dipole radiation above lossy ground, while the third method is a time-domain exact solution for the same problem based on the assumption of perfectly conducting ground. The theoretical background and extent of validity of each technique are reviewed in this thesis. Tlie results derived by applying each method are compared with those obtained using a commercial software package. The time-domain solution for the problem of power transmission line electromagnetic fields is obtained using Fourier and inverse Fourier transforms, rvhich enables us to obtain the electromagnetic waveforms associated with power system transients. The effect of different parameters such as conductivity and relative permittivity of the ground, the line{o-observation point distance, height, and sag of the transmission line is studied. Further, the impact of downscaling of transmission line on different electromagnetic parameters is analyzed, and the appropriate scale factor for each parameter is derived.

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