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.

We consider sensor array imaging with the purpose to image reflectors embedded in a medium. Array imaging consists in two steps. In the first step waves emitted by an array of sources probe the medium to be imaged and are recorded by an array of receivers. In the second step the recorded signals are processed to form an image of the medium. Array imaging in a scattering medium is limited because coherent signals recorded at the receiver array and coming from a reflector to be imaged are weak and...

We compute numerically the minimizers of the Dirichlet energy$$E\left(u\right)=\frac{1}{2}{\int }_{B^2}{\left|\nabla u\right|}^2\mathrm{d}x$$among maps $u:B^2\to S^2$ from the unit disc into the unit sphere that satisfy a boundary condition and a degree condition. We use a Sobolev gradient algorithm for the minimization and we prove that its continuous version preserves the degree. For the discretization of the problem we use continuous $P_1$ finite elements. We propose an original mesh-refining strategy needed to preserve the degree with the discrete version of the algorithm (which is a preconditioned...

We compute numerically the minimizers of the Dirichlet energy $$E\left(u\right)=\frac{1}{2}{\int }_{B^2}{\left|\nabla u\right|}^2\mathrm{d}x$$ among maps $u:B^2\to S^2$ from the unit disc into the unit sphere that satisfy a boundary condition and a degree condition. We use a Sobolev gradient algorithm for the minimization and we prove that its continuous version preserves the degree. For the discretization of the problem we use continuous P1 finite elements. We propose an original mesh-refining strategy needed to preserve the degree with the discrete version of the algorithm (which...

The paper deals with boundary value problems for systems of nonlinear elliptic equations in a relatively general form. Theorems based on monotone operator theory and concerning the existence of weak solutions of such a system, as well as the convergence of discretized problem solutions are presented. As an example, the approach is applied to the stationary Van Roosbroeck’s system, arising in semiconductor device modelling. A convergent algorithm suitable for solving sets of algebraic equations generated...

In this paper, we present a nonlinear model for laser-plasma interaction describing the Raman amplification. This system is a quasilinear coupling of several Zakharov systems. We handle the Cauchy problem and we give some well-posedness and ill-posedness result for some subsystems.

This talk gives a brief review of some recent progress in the asymptotic analysis of short pulse solutions of nonlinear hyperbolic partial differential equations. This includes descriptions on the scales of geometric optics and diffractive geometric optics, and also studies of special situations where pulses passing through focal points can be analysed.