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We study the essential spectrum of operator pencils associated with anisotropic Maxwell equations, with permittivity $varepsilon$, permeability $mu$ and conductivity $sigma$, on finitely connected unbounded domains. The main result is that the essential spectrum of the Maxwell pencil is the union of two sets: namely, the spectrum of the pencil $mathrm{div}((omegavarepsilon + i sigma) abla,cdot,)$, and the essential spectrum of the Maxwell pencil with constant coefficients. We expect the analysis to be of more general interest and to open avenues to investigation of other questions concerning Maxwells and related systems.
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We show existence of solutions to the Poisson equation on Riemannian manifolds with positive essential spectrum, assuming a sharp pointwise decay on the source function. In particular we can allow the Ricci curvature to be unbounded from below. In comparison with previous works, we can deal with a more general setting both on the spectrum and on the curvature bounds.
We derive the discretized Maxwells equations using the discrete variational derivative method (DVDM), calculate the evolution equation of the constraint, and confirm that the equation is satisfied at the discrete level. Numerical simulations showed that the results obtained by the DVDM are superior to those obtained by the Crank-Nicolson scheme. In addition, we study the two types of the discretized Maxwells equations by the DVDM and conclude that if the evolution equation of the constraint is not conserved at the discrete level, then the numerical results are also unstable.
We propose a controllability method for the numerical solution of time-harmonic Maxwells equations in their first-order formulation. By minimizing a quadratic cost functional, which measures the deviation from periodicity, the controllability method determines iteratively a periodic solution in the time domain. At each conjugate gradient iteration, the gradient of the cost functional is simply computed by running any time-dependent simulation code forward and backward for one period, thus leading to a non-intrusive implementation easily integrated into existing software. Moreover, the proposed algorithm automatically inherits the parallelism, scalability, and low memory footprint of the underlying time-domain solver. Since the time-periodic solution obtained by minimization is not necessarily unique, we apply a cheap post-processing filtering procedure which recovers the time-harmonic solution from any minimizer. Finally, we present a series of numerical examples which show that our algorithm greatly speeds up the convergence towards the desired time-harmonic solution when compared to simply running the time-marching code until the time-harmonic regime is eventually reached.
Given a self-adjoint operator $H_0$ and a relatively $H_0$-compact self-adjoint operator $V,$ the functions $r_j(z) = - sigma_j^{-1}(z),$ where $sigma_j(z)$ are eigenvalues of the compact operator $(H_0-z)^{-1}V,$ bear a lot of important information about the pair $H_0$ and $V.$ We call them coupling resonances. In case of rank one (and positive) perturbation $V,$ there is only one coupling resonance function, which is a Herglotz function. This case has been studied in depth in the literature, and appears in different situations, such as Sturm-Liouville theory, random Schrodinger operators, harnomic and spectral analyses, etc. The general case is complicated by the fact that the resonance functions are no longer single valued holomorphic functions, and potentially can have quite an erratic behaviour, typical for infinitely-valued holomorphic functions. Of special interest are those coupling resonance functions $r_z$ which approach a real number $r_{lambda+i0}$ from the interval $[0,1]$ as the spectral parameter $z=lambda+iy$ approaches a point $lambda$ of the essential spectrum, since they are responsible for spectral flow through $lambda$ inside essential spectrum when $H_0$ gets deformed to $H_1 = H_0+V$ via the path $H_0 + rV, r in [0,1].$ In this paper it is shown that if the pair $H_0,$ $V$ satisfies the limiting absorption principle, then the coupling resonance functions are well-behaved near the essential spectrum in the following sense. Let $I$ be an open interval inside the essential spectrum of $H_0$ and $epsilon>0.$ Then there exists a compact subset~$K$ of~$I$ such that $| I setminus K | < epsilon,$ and $K$ has a non-tangential neighbourhood in the upper complex half-plane, such that any coupling resonance function is either single-valued in the neighbourhood, or does not take a real value in the interval $[0,1].$
We introduce concepts of essential numerical range for the linear operator pencil $lambdamapsto A-lambda B$. In contrast to the operator essential numerical range, the pencil essential numerical ranges are, in general, neither convex nor even connected. The new concepts allow us to describe the set of spectral pollution when approximating the operator pencil by projection and truncation methods. Moreover, by transforming the operator eigenvalue problem $Tx=lambda x$ into the pencil problem $BTx=lambda Bx$ for suitable choices of $B$, we can obtain non-convex spectral enclosures for $T$ and, in the study of truncation and projection methods, confine spectral pollution to smaller sets than with hitherto known concepts. We apply the results to various block operator matrices. In particular, Theorem 4.12 presents substantial improvements over previously known results for Dirac operators while Theorem 4.5 excludes spectral pollution for a class of non-selfadjoint Schr{o}dinger operators which it has not been possible to treat with existing methods.
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