Discrete compactness and the approximation of Maxwell's equations in

We analyze the use of edge finite element methods to approximate Maxwell's equations in a bounded cavity. Using the theory of collectively compact operators, we prove -convergence for the source and eigenvalue problems. This is the first proof of convergence of the eigenvalue problem for general edge elements, and it extends and unifies the theory for both problems. The convergence results are based on the discrete compactness property of edge element due to Kikuchi. We extend the original work of Kikuchi by proving that edge elements of all orders possess this property.

     

Regularity theorems for Maxwell's equations

Methods using the theory of distributions and Hilbert space operators have been very powerful in the past to achieve uniqueness and existence results for Maxwell's equations. In this paper conditions are given when such abstract “Hilbert space”-solutions represent differentiable “regular” functions which satisfy Maxwell's equations, boundary conditions, and transmission conditions in the classical sense.     

Crank-nicolsen-Galerkin approximation for Maxwell's equations

The Maxwell equations are formulated as an evolution equation in a suitable chosen Hilbert space involving a densely defined closed skew-Hermitian operator which generates a unitary group. A Crank-Nicolsen-Galerkin approximation is then established and convergence is shown by arguments from the theory of approximation of groups of operators.     

Global superconvergence for Maxwell's equations

In this paper, the global superconvergence is analysed on two schemes (a mixed finite element scheme and a finite element scheme) for Maxwell's equations in . Such a supercovergence analysis is achieved by means of the technique of integral identity (which has been used in the supercovergence analysis for many other equations and schemes) on a rectangular mesh, and then are generalized into more general domains and problems with the variable coefficients. Besides being more direct, our analysis generalizes the results of Monk.

     

A ``nonlinear' proof of Pitt's compactness theorem

Using Stegall's variational principle, we present a simple proof of Pitt's theorem that bounded linear operators from into are compact for .

     

Inverse backscattering problem for Maxwell's equations

In this paper we consider the inverse backscattering problem for Maxwell's equations in a non-magnetic inhomogeneous medium, i.e. the magnetic permeability is a fixed constant. We show that the electric permittivity ε is uniquely determined by the trace of the backscattering kernel S(s, −θ, θ) for all s∈ℝ, θ∈ S ² provided that it is a priori close to a constant. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

A compactness theorem for scalar-flat metrics on manifolds with boundary

Let (M ⁿ , g) be a compact Riemannian manifold with boundary ?M. This paper is concerned with the set of scalar-flat metrics which are in the conformal class of g and have ?M as a constant mean curvature hypersurface. We prove that this set is compact for dimensions n ?? 7 under the generic condition that the trace-free 2nd fundamental form of ?M is nonzero everywhere.     

An effective compactness theorem for Coxeter groups

Through highly non-constructive methods, works by Bestvina, Culler, Feighn, Morgan, Rips, Shalen, and Thurston show that if a finitely presented group does not split over a virtually solvable subgroup, then the space of its discrete and faithful actions on \mathbbHⁿ{\mathbb{H}^n} , modulo conjugation, is compact for all dimensions. Although this implies that the space of hyperbolic structures of such groups has finite diameter, the known methods do not give an explicit bound. We establish such a bound for Coxeter groups. We find that either the group splits over a virtually solvable subgroup or there is a constant C and a point in \mathbbHⁿ{\mathbb{H}^n} that is moved no more than C by any generator. The constant C depends only on the number of generators of the group, and is independent of the relators.     

A remark on Mumford’s compactness theorem

It is shown that a recent compactness theorem for Fuchsian groups, due to Mumford, remains valid for groups containing elliptic and parabolic elements. Work partially supported by the National Science Foundation.     

A parametrized compactness theorem under bounded Ricci curvature

We prove a parametrized compactness theorem on manifolds of bounded Ricci curvature, upper bounded diameter, and lower bounded injectivity radius.     

A mortar spectral element method for 3D Maxwell's equations

^** Email: Tahar.Boulmezaoud{at}univ-pau.fr^*** Email: Mohammed.Elrhabi{at}math.jussieu.fr In this paper we propose a mortar spectral element method forsolving Maxwell's equations in 3D bounded cavities. The methodis based on a non-conforming decomposition of the domain intothe union of non-overlapping parallelepipeds. After provingan error estimate, we present some 3D computational resultswhich confirm the performance of the method.     

An overfilled cavity problem for Maxwell's equations

This paper is concerned with the mathematical analysis of the scattering of a time‐harmonic electromagnetic plane wave by an open and overfilled cavity that is embedded in a perfect electrically conducting infinite ground plane, where the electromagnetic wave propagation is governed by the Maxwell equations. Above the flat ground surface and the open aperture of the cavity, the space is assumed to be filled with a homogeneous medium with a constant permittivity and permeability, whereas the interior of the cavity is filled with some inhomogeneous medium with a variable permittivity and permeability. The scattering problem is modeled as a boundary value problem over a bounded domain, with transparent boundary condition proposed on the hemisphere enclosing the inhomogeneity represented by the cavity. The existence and uniqueness of the weak solution for the model problem are established by using a variational approach. The perfectly matched layer (PML) method is investigated to truncate the unbounded electromagnetic cavity scattering problem. It is shown that the truncated PML problem attains a unique solution. An explicit error estimate is given between the solution of the original scattering problem and that of the truncated PML problem. The error estimate implies that the PML solution converges exponentially to the original cavity scattering problem by increasing either the PML medium parameter or the PML layer thickness. The convergence result is expected to be useful for determining the PML medium parameter in the computational electromagnetic scattering problem. Copyright © 2012 John Wiley & Sons, Ltd.