Global Solutions for the One-Dimensional Vlasov-Maxwell System for Laser-Plasma Interaction

We analyse a reduced 1D Vlasov--Maxwell system introduced recently in the physical literature for studying laser-plasma interaction. This system can be seen as a standard Vlasov equation in which the field is split in two terms: an electrostatic field obtained from Poisson's equation and a vector potential term satisfying a nonlinear wave equation. Both nonlinearities in the Poisson and wave equations are due to the coupling with the Vlasov equation through the charge density. We show global existence of weak solutions in the non-relativistic case, and global existence of characteristic solutions in the quasi-relativistic case. Moreover, these solutions are uniquely characterised as fixed points of a certain operator. We also find a global energy functional for the system allowing us to obtain $L^p$-nonlinear stability of some particular equilibria in the periodic setting

WENO schemes applied to the quasi-relativistic Vlasov--Maxwell model for laser-plasma interaction

In this paper we focus on WENO-based methods for the simulation of the 1D Quasi-Relativistic Vlasov--Maxwell (QRVM) model used to describe how a laser wave interacts with and heats a plasma by penetrating into it. We propose several non-oscillatory methods based on either Runge--Kutta (explicit) or Time-Splitting (implicit) time discretizations. We then show preliminary numerical experiments

The Fourier Singular Complement Method for the Poisson problem. Part I: prismatic domains

This is the first part of a threefold article, aimed at solving numerically the Poisson problem in three-dimensional prismatic or axisymmetric domains. In this first part, the Fourier Singular Complement Method is introduced and analysed, in prismatic domains. In the second part, the FSCM is studied in axisymmetric domains with conical vertices, whereas, in the third part, implementation issues, numerical tests and comparisons with other methods are carried out. The method is based on a Fourier expansion in the direction parallel to the reentrant edges of the domain, and on an improved variant of the Singular Complement Method in the 2D section perpendicular to those edges. Neither refinements near the reentrant edges of the domain nor cut-off functions are required in the computations to achieve an optimal convergence order in terms of the mesh size and the number of Fourier modes used

The Fourier Singular Complement Method for the Poisson problem. Part II: axisymmetric domains

This paper is the second part of a threefold article, aimed at solving numerically the Poisson problem in three-dimensional prismatic or axisymmetric domains. In the first part of this series, the Fourier Singular Complement Method was introduced and analysed, in prismatic domains. In this second part, the FSCM is studied in axisymmetric domains with conical vertices, whereas, in the third part, implementation issues, numerical tests and comparisons with other methods are carried out. The method is based on a Fourier expansion in the direction parallel to the reentrant edges of the domain, and on an improved variant of the Singular Complement Method in the 2D section perpendicular to those edges. Neither refinements near the reentrant edges or vertices of the domain, nor cut-off functions are required in the computations to achieve an optimal convergence order in terms of the mesh size and the number of Fourier modes used

Polynomial bounds for the solutions of parametric transmission problems on smooth, bounded domains

We consider a \emph{family} $(P_\omega)_{\omega \in \Omega}$ of elliptic second order differential operators on a domain $U_0 \subset \mathbb{R}^m$ whose coefficients depend on the space variable $x \in U_0$ and on $\omega \in \Omega,$ a probability space. We allow the coefficients $a_{ij}$ of $P_\omega$ to have jumps over a fixed interface $\Gamma \subset U_0$ (independent of $\omega \in \Omega$). We obtain polynomial in the norms of the coefficients estimates on the norm of the solution $u_\omega$ to the equation $P_\omega u_\omega = f$ with transmission and mixed boundary conditions (we consider ``sign-changing'' problems as well). In particular, we show that, if $f$ and the coefficients $a_{ij}$ are smooth enough and follow a log-normal-type distribution, then the map $\Omega \ni \omega \to \|u_\omega\|_{H^{k+1}(U_0)}$ is in $L^p(\Omega)$, for all $1 \le p < \infty$. The same is true for the norms of the inverses of the resulting operators. We expect our estimates to be useful in Uncertainty Quantification.Comment: We fixed a small .tex problem in the abstract on the site (the manuscript has not changed

Electromagnetic wave propagation and absorption in magnetised plasmas: variational formulations and domain decomposition

We consider a model for the propagation and absorption of electromagnetic waves (in the time-harmonic regime) in a magnetised plasma. We present a rigorous derivation of the model and several boundary conditions modelling wave injection into the plasma. Then we propose several variational formulations, mixed and non-mixed, and prove their well-posedness thanks to a theorem by S\'ebelin et~al. Finally, we propose a non-overlapping domain decomposition framework, show its well-posedness and equivalence with the one-domain formulation. These results appear strongly linked to the spectral properties of the plasma dielectric tensor

Tout ce que vous avez toujours voulu savoir sur Maxwell sans jamais oser le demander

Nous présentons quelques aspects de la théorie mathématique et de la résolution numérique des équations de Maxwell instationnaires, plus particulièrement en vue de la simulation de particules chargées. Deux thématiques servent de fil conducteur à cet article: le traitement la contrainte de divergence et celui des singularités. Les équations habituelles, du premier ordre, peuvent être reformulées en équations du second ordre, plus adaptées à la simulation par éléments finis. On peut d'autre part intégrer différents traitements de la contrainte de divergence, même en présence de données bruitées. Nous esquissons la preuve d'existence et d'unicité de la solution de ces équations. La régularité de cette solution dépend fortement des singularités du domaine de calcul, et influe à son tour sur le choix de la méthode d'éléments finis. Deux méthodes sont examinées en détail: les éléments d'arête et les éléments nodaux; pour ces derniers, deux variantes permettent une prise en compte efficace des singularités. Nous donnons des estimations d'erreurs optimales pour toutes les variantes

Développement asymptotique et approximation de la solution des équations de Maxwell dans un polygone

We present an improved version of the Singular Complement Method (SCM) for Maxwell's equations, which relies on an asymptotic expansion of the solution near non-regular points. This method allows to recover an optimal error estimate when used with $\mathbb{P}_1$ Lagrange finite elements; extension to higher-degree elements is possible. It can be applied to static, harmonic, or time-dependent problems