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In this paper we propose a time-domain finite element method for modeling of electromagnetic cloaks. The permittivity and permeability of the cloak model are described by the Drude dispersion model. The model to be solved is quite challenging in that we have to solve a coupled problem with different partial differential equations given in different regions. Our method is based on a mixed finite element method using edge elements with different types of meshes in different regions. Numerical results demonstrate that our algorithm is quite effective for simulating cloaks in time-domain. To our knowledge, this is the first cloak simulation carried out by the time-domain finite element method.

Previous work has shown that in a two-dimensional periodic medium under focusing or defocusing cubic nonlinearities, gap solitons in the form of low-amplitude and slowly modulated single-Bloch-wave packets can bifurcate out from the edges of Bloch bands. In this paper, linear stability properties of these gap solitons near band edges are determined both analytically and numerically. Through asymptotic analysis, it is shown that these gap solitons are linearly unstable if the slope of their power curve at the band edge has the opposite sign of nonlinearity
here focusing nonlinearity is said to have a positive sign, and defocusing nonlinearity to have a negative sign. An equivalent condition for linear instability is that the power of the gap solitons near the band edge is lower than the limit power value on the band edge. Through numerical computations of the power curves, it is found that this condition is always satisfied, thus two-dimensional gap solitons near band edges are linearly unstable. The analytical formula for the unstable eigenvalue of gap solitons near band edges is also asymptotically derived. It is shown that this unstable eigenvalue is proportional to the cubic power of the soliton’s amplitude, and it induces width instabilities of gap solitons. A comparison between this analytical eigenvalue formula and numerically computed eigenvalues shows excellent agreement.

In this paper, we propose a novel and simple technique to construct effective difference schemes for solving systems of singularly perturbed convection-diffusionreaction equations, whose solutions may display boundary or interior layers. We illustrate the technique by taking the Il’in-Allen-Southwell scheme for 1-D scalar equations as a basis to derive a formally second-order scheme for 1-D coupled systems and then extend the scheme to 2-D case by employing an alternating direction approach. Numerical examples are given to demonstrate the high performance of the obtained scheme on uniform meshes as well as piecewise-uniform Shishkin meshes.