In this paper we are interested in the numerical modeling of absorbing ferromagnetic materials obeying the non-linear Landau-Lifchitz-Gilbert law with respect to the propagation and scattering of electromagnetic waves. In this work we consider the 1D problem. We first show that the corresponding Cauchy problem has a unique global solution. We then derive a numerical scheme based on an appropriate modification of Yee's scheme, that we show to preserve some important properties of the continuous...

Using Maxwell’s mental imagery of a tube of fluid motion of an imaginary fluid, we derive his equations $curl𝐄=-\frac{\partial 𝐁}{\partial t}$, $curl𝐇=\frac{\partial 𝐃}{\partial t}+𝐣$, $div𝐃=\varrho$, $div𝐁=0$, which together with the constituting relations $𝐃={\epsilon }_0𝐄$, $𝐁={\mu }_0𝐇$, form what we call today Maxwell’s equations. Main tools are the divergence, curl and gradient integration theorems and a version of Poincare’s lemma formulated in vector calculus notation. Remarks on the history of the development of electrodynamic theory, quotations and references to original and secondary literature complement...

We present and analyze an interior penalty method for the numerical discretization of the indefinite time-harmonic Maxwell equations in mixed form. The method is based on the mixed discretization of the curl-curl operator developed in [Houston et al., J. Sci. Comp. 22 (2005) 325–356] and can be understood as a non-stabilized variant of the approach proposed in [Perugia et al., Comput. Methods Appl. Mech. Engrg. 191 (2002) 4675–4697]. We show the well-posedness of this approach and derive optimal...

We present and analyze an interior penalty method for the numerical discretization of the indefinite time-harmonic Maxwell equations in mixed form. The method is based on the mixed discretization of the curl-curl operator developed in [Houston et al., J. Sci. Comp.22 (2005) 325–356] and can be understood as a non-stabilized variant of the approach proposed in [Perugia et al., Comput. Methods Appl. Mech. Engrg.191 (2002) 4675–4697]. We show the well-posedness of this approach and derive optimal...

Nous étudions l’effet d’une couche mince rugueuse périodique déposée sur une structure semi-infinie, dans le contexte Helmholtz bi-dimensionnel. Formellement, nous obtenons des conditions de transmission équivalentes à l’ordre 1, par des techniques de type homogénéisation. Suivent alors la résolution du problème du milieu effectif éclairé par une onde plane, et le calcul de la fonction de Green effective ; le tout par analyse de Fourier. Dans un deuxième temps, nous considérons le problème de diffraction...

In this work we consider the magnetic NLS equation$${(\frac{\hslash }{i}\nabla -A\left(x\right))}^{2}u+V\left(x\right)u-{f\left(\right|u|}^2)u=0\text{in}{ℝ}^N(0.1)$$where $N\ge 3$, $A:{ℝ}^N\to {ℝ}^N$ is a magnetic potential, possibly unbounded, $V:{ℝ}^N\to ℝ$ is a multi-well electric potential, which can vanish somewhere, $f$ is a subcritical nonlinear term. We prove the existence of a semiclassical multi-peak solution $u:{ℝ}^N\to ℂ$ to (0.1), under conditions on the nonlinearity which are nearly optimal.

In this work we consider the magnetic NLS equation $${(\frac{\hslash }{i}\nabla -A\left(x\right))}^{2}u+V\left(x\right)u-{f\left(\right|u|}^2)u=0\text{in}{ℝ}^N(0.1)$$ where $N\ge 3$, $A:{ℝ}^N\to {ℝ}^N$ is a magnetic potential, possibly unbounded, $V:{ℝ}^N\to ℝ$ is a multi-well electric potential, which can vanish somewhere, f is a subcritical nonlinear term. We prove the existence of a semiclassical multi-peak solution $u:{ℝ}^N\to ℂ$ to (0.1), under conditions on the nonlinearity which are nearly optimal.