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Simon Labrunie

Publications and source records attributed to Simon Labrunie.

12 recordsLinked to original sources
arXiv2024

Uniform estimates for a family of Poisson problems: `rounding off the corners'

We prove \emph{uniform solvability estimates} for certain families of elliptic problems posed in a bounded family of domains (for example, a sequence that converges to another domain). We provide uniform estimates both in weighted and in usual Sobolev spaces. When the limit domain is a \emph{polygon} and the other domains are smooth, our results amount to rounding off'' the corners of the limit domain. The technique of proof is based on a suitable conformal modification of the metric, which makes the union of the domains a manifold with boundary and relative bounded geometry.

math.AP↗
arXiv2023

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.

math.AP↗
arXiv2020

Linearised electrodynamics and stabilisation of a cold magnetised plasma

We consider a linearized Euler--Maxwell model for the propagation and absorption of electromagnetic waves in a magnetized plasma. We present the derivation of the model, and we show its well-posedeness, its strong and polynomial stability under suitable and fairly general assumptions, its exponential stability in the same conditions as the Maxwell system, and finally its convergence to the time-harmonic regime. No homogeneity assumption is made, and the topological and geometrical assumptions on the domain are minimal. These results appear strongly linked to the spectral properties of various matrices describing the anisotropy and other plasma properties.

math.AP↗
arXiv2015

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.

math.AP↗
arXiv2014

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.

math.AP↗
arXiv2006

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.

math.AP↗
arXiv2006

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.

math.AP↗
arXiv2006

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.

math.AP↗
arXiv2006

On the harmonic Boltzmannian waves in laser-plasma interaction

We study the permanent regimes of the reduced Vlasov-Maxwell system for laser-plasma interaction. A non-relativistic and two different relativistic models are investigated. We prove the existence of solutions where the distribution function is Boltzmannian and the electromagnetic variables are time-harmonic and circularly polarized.

math.AP↗
arXiv1996

Discrete version of the Chazy class III equation

We study the discretisation of the Chazy class III equation by two means: a discrete Painlevé test, and the preservation of a two-parameter solution to the continuous equation. We get that way a best discretisation scheme.

solv-int↗
arXiv1996

A geometrical method towards first integrals for dynamical systems

We develop a method, based on Darboux' and Liouville's works, to find first integrals and/or invariant manifolds for a physically relevant class of dynamical systems, without making any assumption on these elements' form. We apply it to three dynamical systems: Lotka--Volterra, Lorenz and Rikitake.

solv-int↗