Interior penalty DG methods for Maxwell’s equations in dispersive media

In this paper, we develop a fully-discrete interior penalty discontinuous Galerkin method for solving the time-dependent Maxwell’s equations in dispersive media. The model is described by a vector integral–differential equation. Our scheme is proved to be unconditionally stable and achieve optimal error estimates in both L² norm and energy norm. The scheme is implemented and numerical results supporting our analysis are presented.     

Non-conformal domain decomposition method with second-order transmission conditions for time-harmonic electromagnetics

Non-overlapping domain decomposition (DD) methods provide efficient algorithms for solving time-harmonic Maxwell equations. It has been shown that the convergence of DD algorithms can be improved significantly by using high order transmission conditions. In this paper, we extend a newly developed second-order transmission condition (SOTC), which involves two second-order transverse derivatives, to facilitate fast convergence in the non-conformal DD algorithms. However, the non-conformal nature of the DD methods introduces an additional technical difficulty, which results in poor convergence in many real-life applications. To mitigate the difficulty, a corner-edge penalty method is proposed and implemented in conjunction with the SOTC to obtain truly robust solver performance. Numerical results verify the analysis and demonstrate the effectiveness of the proposed methods on a few model problems. Finally, drastically improved convergence, compared to the conventional Robin transmission condition, was observed for an electrically large problem of practical interest.     

The Discontinuous Galerkin Time-Domain method for Maxwell’s equations with anisotropic materials

The Discontinuous Galerkin method is an accurate and efficient way to numerically solve the time-dependent Maxwell equations. In this paper, we extend the basic, two-dimensional formulation for isotropic materials to allow anisotropic permittivity tensors. Using a reference system with an analytical solution, we demonstrate that our extensions do not alter the superior convergence characteristics of the fundamental algorithm. We further apply our method to cylindrical invisibility cloaks to investigate the performance which can be achieved in experiments.     

A linear nonconforming finite element method for Maxwell’s equations in two dimensions. Part I: Frequency domain

We suggest a linear nonconforming triangular element for Maxwell’s equations and test it in the context of the vector Helmholtz equation. The element uses discontinuous normal fields and tangential fields with continuity at the midpoint of the element sides, an approximation related to the Crouzeix–Raviart element for Stokes. The element is stabilized using the jump of the tangential fields, giving us a free parameter to decide. We give dispersion relations for different stability parameters and give some numerical examples, where the results converge quadratically with the mesh size for problems with smooth boundaries. The proposed element is free from spurious solutions and, for cavity eigenvalue problems, the eigenfrequencies that correspond to well-resolved eigenmodes are reproduced with the correct multiplicity.     

Effective coefficients of quasi-steady Maxwell’s equations with multiscale isotropic random conductivity

The effective coefficients in the quasi-steady Maxwell’s equations are calculated for a multiscale isotropic medium by using a subgrid modeling approach. The conductivity is mathematically represented by a Kolmogorov multiplicative continuous cascade with a lognormal probability distribution. The scale of the solution domain is assumed to be large as compared with the scale of heterogeneities of the medium. The theoretical results obtained in the paper are compared with the results of a direct 3D numerical simulation and the results of the conventional perturbation theory.     

Sufficient uniqueness conditions for the solution of the time harmonic Maxwell’s equations associated with surface impedance boundary conditions

We consider the time-harmonic Maxwell’s equations for the scattering or radiating problem from a 3-D object that is either a metallic surface coated with material layers (MCS) or a dichroic structure (DS) made up of multiple frequency selective surfaces (FSS) embedded in materials. Low or high order impedance boundary conditions (IBC) are employed to reduce the numerical complexity of the solution of this problem via an integral equation or a finite element formulation. An IBC links the tangential components of the electric field to those of the magnetic field on the outer surface of the MCS, or on the FSSs, and avoids the solution of Maxwell’s equations inside the inhomogeneous domain for a MCS or, for a DS, the meshing of the numerous unit cells in a FSS. Sufficient uniqueness conditions (SUC) are established for the solutions of Maxwell’s equations associated with these IBCs, the performances of which, when constrained by the corresponding SUCs, are numerically evaluated for an infinite or finite planar structure.     

Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations

The technique of applying form-invariant, spatial coordinate transformations of Maxwell’s equations can facilitate the design of structures with unique electromagnetic or optical functionality. Here, we illustrate the transformation-optical approach in the designs of a square electromagnetic cloak and an omni-directional electromagnetic field concentrator. The transformation equations are described and the functionality of the devices is numerically confirmed by two-dimensional finite element simulations. The two devices presented demonstrate that the transformation optic approach leads to the specification of complex, anisotropic and inhomogeneous materials with well directed and distinct electromagnetic behavior.     

Surface electromagnetic waves supported by nano conducting layers with inhomogeneities in the conductivity profile

Conducting interfaces and nano conducting layers can support surface electromagnetic waves. Uniform charge layers of non-zero thickness and their asymptotic behavior toward conducting interfaces of infinitely small thicknesses, where the thin charge layer is modeled via a surface conductivity σ _s , are already studied. Here, the possible effects of inhomogeneity in the conductivity profile of the thin conducting layers are investigated for the first time and a new approximate yet accurate enough analytical formulation for mode extraction in such structures is given. In order to rigorously analyze the structure and justify the proposed approximate formulation, the Galerkin’s method with Legendre polynomial basis functions is applied, i.e. the transverse electric field for the TE polarized surface waves and the transverse magnetic field for the TM polarized surface waves are each expanded in terms of Legendre polynomials and then each eigenmode; subjected to appropriate boundary conditions, is sought in the complete space spanned by Legendre basis functions. The proposed approximate solution is then proved to be accurate. In particular, sinusoidal fluctuations are introduced into formerly uniform conductivity profiles and it is numerically demonstrated that surface electromagnetic waves supported by nano conducting layers are not much sensitive to the very shape of conductivity profiles.     

A high-order numerical method for the nonlinear Helmholtz equation in multidimensional layered media

We present a novel computational methodology for solving the scalar nonlinear Helmholtz equation (NLH) that governs the propagation of laser light in Kerr dielectrics.