arXiv subjects

Xavier Claeys

Publications and source records attributed to Xavier Claeys.

25 records · Page 2Linked to original sources

arXiv2021

Analysis of the SORAS domain decomposition preconditioner for non-self-adjoint or indefinite problems

We analyze the convergence of the one-level overlapping domain decomposition preconditioner SORAS (Symmetrized Optimized Restricted Additive Schwarz) applied to a generic linear system whose matrix is not necessarily symmetric/self-adjoint nor positive definite. By generalizing the theory for the Helmholtz equation developed in [I.G. Graham, E.A. Spence, and J. Zou, SIAM J.Numer.Anal., 2020], we identify a list of assumptions and estimates that are sufficient to obtain an upper bound on the norm of the preconditioned matrix, and a lower bound on the distance of its field of values from the origin. We stress that our theory is general in the sense that it is not specific to one particular boundary value problem. Moreover, it does not rely on a coarse mesh whose elements are sufficiently small. As an illustration of this framework, we prove new estimates for overlapping domain decomposition methods with Robin-type transmission conditions for the heterogeneous reaction-convection-diffusion equation (to prove the stability assumption for this equation we consider the case of a coercive bilinear form, which is non-symmetric, though).

math.NA↗

arXiv2019

A new variant of the Optimised Schwarz Method for arbitrary non-overlapping subdomain partitions

We consider a scalar wave propagation in harmonic regime modelled by Helmholtz equation with heterogeneous coefficients. Using the Multi-Trace Formalism (MTF), we propose a new variant of the Optimized Schwarz Method (OSM) that can accomodate the presence of cross-points in the subdomain partition. This leads to the derivation of a strongly coercive formulation of our Helmholtz problem posed on the union of all interfaces. The corresponding operator takes the form "identity + contraction".

math.AP↗

arXiv2017

Oscillating behaviour of the spectrum for a plasmonic problem in a domain with a rounded corner

We investigate the eigenvalue problem $-\text{div}(σ\nabla u) = λu\ (\mathscr{P})$ in a 2D domain $Ω$ divided into two regions $Ω_{\pm}$. We are interested in situations where $σ$ takes positive values on $Ω_{+}$ and negative ones on $Ω_{-}$. Such problems appear in time harmonic electromagnetics in the modeling of plasmonic technologies. In a recent work [15], we highlighted an unusual instability phenomenon for the source term problem associated with $(\mathscr{P})$: for certain configurations, when the interface between the subdomains $Ω_{\pm}$ presents a rounded corner, the solution may depend critically on the value of the rounding parameter. In the present article, we explain this property studying the eigenvalue problem $(\mathscr{P})$. We provide an asymptotic expansion of the eigenvalues and prove error estimates. We establish an oscillatory behaviour of the eigenvalues as the rounding parameter of the corner tends to zero. We end the paper illustrating this phenomenon with numerical experiments.

math.AP↗

arXiv2015

Essential spectrum of local multi-trace boundary integral operators

Considering pure transmission scattering problems in piecewise constant media, we derive an exact analytic formula for the spectrum of the corresponding local multi-trace boundary integral operators in the case where the geometrical configuration does not involve any junction point and all wave numbers equal. We deduce from this the essential spectrum in the case where wave numbers vary. Numerical evidences of these theoretical results are also presented.

math.AP↗

arXiv2015

Spectrum of a diffusion operator with coefficient changing sign over a small inclusion

We study a spectral problem $(\mathscr{P}^δ)$ for a diffusion like equation in a 3D domain $Ω$. The main originality lies in the presence of a parameter $σ^δ$, whose sign changes on $Ω$, in the principal part of the operator we consider. More precisely, $σ^δ$ is positive on $Ω$ except in a small inclusion of size $δ>0$. Because of the sign-change of $σ^δ$, for all $δ>0$ the spectrum of $(\mathscr{P}^δ)$ consists of two sequences converging to $\pm\infty$. However, at the limit $δ=0$, the small inclusion vanishes so that there should only remain positive spectrum for $(\mathscr{P}^δ)$. What happens to the negative spectrum? In this paper, we prove that the positive spectrum of $(\mathscr{P}^δ)$ tends to the spectrum of the problem without the small inclusion. On the other hand, we establish that each negative eigenvalue of $(\mathscr{P}^δ)$ behaves like $δ^{-2}μ$ for some constant $μ<0$. We also show that the eigenfunctions associated with the negative eigenvalues are localized around the small inclusion. We end the article providing 2D numerical experiments illustrating these results.

math.AP↗

arXiv2014

A numerical approach for the Poisson equation in a planar domain with a small inclusion

We consider the Poisson equation in a domain with a small hole of size $δ$. We present a simple numerical method, based on an asymptotic analysis, which allows to approximate robustly the far field of the solution as $δ$ goes to zero without meshing the small hole. We prove the stability of the scheme and provide error estimates. We end the paper with numerical experiments illustrating the efficiency of the technique.

math.NA↗

arXiv2013

A curious instability phenomenon for a rounded corner in presence of a negative material

We study a 2D scalar harmonic wave transmission problem between a classical dielectric and a medium with a real-valued negative permittivity/permeability which models a metal at optical frequency or an ideal negative metamaterial. We highlight an unusual instability phenomenon for this problem when the interface between the two media presents a rounded corner. To establish this result, we provide an asymptotic expansion of the solution, when it is well-defined, in the geometry with a rounded corner. Then, we prove error estimates. Finally, a careful study of the asymptotic expansion allows us to conclude that the solution, when it is well-defined, depends critically on the value of the rounding parameter. We end the paper with a numerical illustration of this instability phenomenon.

math.AP↗