Characterization of the singular part of the solution of Maxwell's equations in a polyhedral domain

The solution of Maxwell's equations in a non‐convex polyhedral domain is less regular than in a smooth or convex polyhedral domain. In this paper we show that this solution can be decomposed into the orthogonal sum of a singular part and a regular part, and we give a characterization of the singular part. We also prove that the decomposition is linked to the one associated to the scalar Laplacian. Copyright © 1999 John Wiley & Sons, Ltd.     

Convergence of the mixed finite element method for Maxwell's equations with nonlinear conductivity

In this paper, we study a numerical scheme to solve coupled Maxwell's equations with a nonlinear conductivity. This model plays an important role in the study of type‐II superconductors. The approximation scheme is based on backward Euler discretization in time and mixed conforming finite elements in space. We will prove convergence of this scheme to the unique weak solution of the problem and develop the corresponding error estimates. As a next step, we study the stability of the scheme in the quasi‐static limit ? → 0 and present the corresponding convergence rate. Finally, we support the theory by several numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.     

Transmission problems for Maxwell's equations with weakly Lipschitz interfaces

We prove sufficient conditions on material constants, frequency and Lipschitz regularity of interface for well posedness of a generalized Maxwell transmission problem in finite energy norms. This is done by embedding Maxwell's equations in an elliptic Dirac equation, by constructing the natural trace space for the transmission problem and using Hodge decompositions for operators d and δ on weakly Lipschitz domains to prove stability. We also obtain results for boundary value problems and transmission problems for the Hodge–Dirac equation and prove spectral estimates for boundary singular integral operators related to double layer potentials. Copyright © 2005 John Wiley & Sons, Ltd.     

On singularities of solution of Maxwell's equations in axisymmetric domains with conical points

Boundary value problems (BVP) in three‐dimensional axisymmetric domains can be treated more efficiently by partial Fourier analysis. Partial Fourier analysis is applied to time‐harmonic Maxwell's equations in three‐dimensional axisymmetric domains with conical points on the rotation axis thereby reducing the three dimensional BVP to an infinite sequence of 2D BVPs on the plane meridian domain Ω_a?? of . The regularity of the solutions u _n (n∈?₀:={0, 1, 2,…}) of the two dimensional BVPs is investigated and it is proved that the asymptotic behaviour of the solutions u _n near an angular point on the rotation axis can be characterized by singularity functions related to the solutions of some associated Legendre equations. By means of numerical experiments, it is shown that the solutions u _n for n∈?₀\{1} belong to the Sobolev space H² irrespective of the size of the solid angle at the conical point. However, the regularity of the coefficient u ₁ depends on the size of the solid angle at the conical point. The singular solutions of the three dimensional BVP are obtained by Fourier synthesis. Copyright © 2006 John Wiley & Sons, Ltd.     

Analysis of the energy‐conserved S‐FDTD scheme for variable coefficient Maxwell's equations in disk domains

In this paper, we analyze the energy‐conserved splitting finite‐difference time‐domain (FDTD) scheme for variable coefficient Maxwell's equations in two‐dimensional disk domains. The approach is energy‐conserved, unconditionally stable, and effective. We strictly prove that the EC‐S‐FDTD scheme for the variable coefficient Maxwell's equations in disk domains is of second order accuracy both in time and space. It is also strictly proved that the scheme is energy‐conserved, and the discrete divergence‐free is of second order convergence. Numerical experiments confirm the theoretical results, and practical test is simulated as well to demonstrate the efficiency of the proposed EC‐S‐FDTD scheme. Copyright © 2015 John Wiley & Sons, Ltd.     

On the solution of Maxwell's equations in polygonal domains

This paper is concerned with the structure of the singular and regular parts of the solution of time‐harmonic Maxwell's equations in polygonal plane domains and their effective numerical treatment. The asymptotic behaviour of the solution near corner points of the domain is studied by means of discrete Fourier transformation and it is proved that the solution of the boundary value problem does not belong locally to H² when the boundary of the domain has non‐acute angles. A splitting of the solution into a regular part belonging to the space H², and an explicitly described singular part is presented. For the numerical treatment of the boundary value problem, we propose a finite element discretization which combines local mesh grading and the singular field methods and derive a priori error estimates that show optimal convergence as known for the classical finite element method for problems with regular solutions. Copyright © 2006 John Wiley & Sons, Ltd.     

New energy identities and super convergence analysis of the energy conserved splitting FDTD methods for 3D Maxwell's equations

The energy‐conserved splitting finite‐difference time‐domain (EC‐S‐FDTD) method has recently been proposed to solve the Maxwell equations with second order accuracy while numerically keep the L² energy conservation laws of the equations. In this paper, the EC‐S‐FDTD scheme for the 3D Maxwell equations is proved to be energy‐conserved and unconditionally stable in the discrete H¹ norm. The EC‐S‐FDTD scheme is of second‐order accuracy both in time step and spatial steps, which suggests the super‐convergence of this scheme in the discrete H¹ norm. And the divergence of the electric field of the EC‐S‐FDTD scheme in the discrete L² norm is second‐order accurate. Numerical experiments confirm our theoretical analysis. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

An enhanced finite difference time domain method for two dimensional Maxwell's equations

An enhanced finite-difference time-domain (FDTD) algorithm is built to solve the transverse electric two-dimensional Maxwell's equations with inhomogeneous dielectric media where the electric fields are discontinuous across the dielectric interface. The new algorithm is derived based upon the integral version of the Maxwell's equations as well as the relationship between the electric fields across the interface. To resolve the instability issue of Yee's scheme (staircasing) caused by discontinuous permittivity across the interface, our algorithm revises the permittivities and makes some corrections to the scheme for the cells around the interface. It is also an improvement over the contour-path effective permittivity algorithm by including some extra terms in the formulas. The scheme is validated in solving the scattering of a dielectric cylinder with exact solution from Mie theory and is then compared with the above contour-path method, the usual staircasing and the volume-average method. The numerical results demonstrate that the new algorithm has achieved significant improvement in accuracy over other methods. Furthermore, the algorithm has a simple structure and can be merged into current FDTD software packages easily. The C++ source code for this paper is provided as supporting information for public access.     

Boundary element approximation for Maxwell's eigenvalue problem

We introduce a new method for computing eigenvalues of the Maxwell operator with boundary finite elements. On bounded domains with piecewise constant material coefficients, the Maxwell solution for fixed wave number can be represented by boundary integrals, which allows to reduce the eigenvalue problem to a nonlinear problem for determining the wave number along with boundary and interface traces. A Galerkin discretization yields a smooth nonlinear matrix eigenvalue problem that is solved by Newton's method or, alternatively, the contour integral method. Several numerical results including an application to the band structure computation of a photonic crystal illustrate the efficiency of this approach. Copyright © 2013 John Wiley & Sons, Ltd.     

Superconvergence analysis for time‐dependent Maxwell's equations in metamaterials

In this article, we consider the time‐dependent Maxwell's equations modeling wave propagation in metamaterials. One‐order higher global superclose results in the L² norm are proved for several semidiscrete and fully discrete schemes developed for solving this model using nonuniform cubic and rectangular edge elements. Furthermore, L^∞ superconvergence at element centers is proved for the lowest order rectangular edge element. To our best knowledge, such pointwise superconvergence result and its proof are original, and we are unaware of any other publications on this issue. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential 2011     

An inverse problem for Maxwell's equations in isotropic and non-stationary media

In this article, we consider Maxwell's equations in an isotropic, inhomogeneous and non-stationary medium. We discuss an inverse problem of determining the t-independent components of the coefficients ?, μ in the constitutive relations from a finite number of interior measurements. We prove a Lipschitz stability estimate for the inverse problem by applying the argument on the basis of Carleman estimate.     

Finite element study of time‐dependent Maxwell's equations in dispersive media

In this article, we consider the time‐dependent Maxwell's equations in a bounded domain when dispersive media are involved. The Crank‐Nicolson scheme is developed to approximate the electric field equation by Nedelec edge elements and is proved to be optimal convergent in energy norm. The analysis is carried out for Debye medium, but the same results hold true for other dispersive media such as plasma and Lorentz medium. Furthermore, our analysis extends straightforward to cases when a dispersive medium and a simple medium (such as air) are coupled. Mathematics Subject Classification (2000): 65N30, 35L15, 78‐08. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Two-dimensional Maxwell's equations with sign-changing coefficients

We consider the theoretical study of time harmonic Maxwell's equations in presence of sign-changing coefficients, in a two-dimensional configuration. Classically, the problems for both the Transverse Magnetic and the Transverse Electric polarizations reduce to an equivalent scalar Helmholtz type equation. For this scalar equation, we have already studied consequences of the presence of sign-changing coefficients in previous papers, and we summarize here the main results. Then we focus on the alternative approach which relies on the two-dimensional vectorial formulations of the TM or TE problems, and we exhibit some unexpected effects of the sign-change of the coefficients. In the process, we provide new results on the scalar equations.     

Maxwell's equations in an unbounded structure

This paper is concerned with the mathematical analysis of the electromagnetic wave scattering by an unbounded dielectric medium, which is mounted on a perfectly conducting infinite plane. By introducing a transparent boundary condition on a plane surface confining the medium, the scattering problem is modeled as a boundary value problem of Maxwell's equations. Based on a variational formulation, the problem is shown to have a unique weak solution for a wide class of dielectric permittivity and magnetic permeability by using the generalized Lax–Milgram theorem. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical solution of an inverse initial boundary-value problem for the full time-dependent Maxwell's equations in the presence of imperfections of small volume

We consider the numerical solution, in a three-dimensional bounded domain, of the inverse problem for identifying the location of small electromagnetic imperfections in a medium with homogeneous background. Our numerical algorithm is based on the coupling of a discontinuous Galerkin method for the time-dependent Maxwell's equations, on the exact controllability method and on a Fourier inversion. Several numerical results are given with one and two imperfections and the robustness and accuracy of the numerical method used for the dynamic detection problem are shown.     

A volume integral equation method for periodic scattering problems for anisotropic Maxwell's equations

This paper presents a volume integral equation method for an electromagnetic scattering problem for three-dimensional Maxwell's equations in the presence of a biperiodic, anisotropic, and possibly discontinuous dielectric scatterer. Such scattering problem can be reformulated as a strongly singular volume integral equation (i.e., integral operators that fail to be weakly singular). In this paper, we firstly prove that the strongly singular volume integral equation satisfies a Gårding-type estimate in standard Sobolev spaces. Secondly, we rigorously analyze a spectral Galerkin method for solving the scattering problem. This method relies on the periodization technique of Gennadi Vainikko that allows us to efficiently evaluate the periodized integral operators on trigonometric polynomials using the fast Fourier transform (FFT). The main advantage of the method is its simple implementation that avoids for instance the need to compute quasiperiodic Green's functions. We prove that the numerical solution of the spectral Galerkin method applied to the periodized integral equation converges quasioptimally to the solution of the scattering problem. Some numerical examples are provided for examining the performance of the method.     

Development and analysis of Crank‐Nicolson scheme for metamaterial Maxwell's equations on nonuniform rectangular grids

In this paper, we develop a fully implicit scheme on staggered grids to solve the Maxwell's equations when Drude metamaterial is involved. Unconditional stability and optimal error estimate of the scheme are proved. Numerical results are provided to support the theoretical analysis, and used to demonstrate the applicability of the scheme to simulate the complicated backward wave propagation phenomenon occurring in metamaterials.