Publications

2022

• A mathematical study of a hyperbolic metamaterial in free space
  
  • Ciarlet Patrick
  • Kachanovska Maryna
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2022, 54 (2), pp.2216-2250. Wave propagation in hyperbolic metamaterials is described by the Maxwell equations with a frequency dependent tensor of dielectric permittivity, whose eigenvalues are of different signs. In this case the problem becomes hyperbolic (Klein-Gordon equation) for a certain range of frequencies. The principal theoretical and numerical difficulty comes from the fact that this hyperbolic equation is posed in a free space, without initial conditions provided. The subject of the work is the theoretical justification of this problem. In particular, this includes the construction of a radiation condition, a well-posedness result, a limiting absorption principle and regularity estimates on the solution. (10.1137/21M1404223)
  DOI : 10.1137/21M1404223
• Local transparent boundary conditions for wave propagation in fractal trees (ii): error and complexity analysis
  
  • Joly Patrick
  • Kachanovska Maryna
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2022, 60 (2). This work is dedicated to a refined error analysis of the high-order transparent boundary conditions introduced in the companion work [8] for the weighted wave equation on a fractal tree. The construction of such boundary conditions relies on truncating the meromorphic series that represents the symbol of the Dirichlet-to-Neumann operator. The error induced by the truncation depends on the behaviour of the eigenvalues and the eigenfunctions of the weighted Laplacian on a self-similar metric tree. In this work we quantify this error by computing asymptotics of the eigenvalues and bounds for Neumann traces of the eigenfunctions. We prove the sharpness of the obtained bounds for a class of self-similar trees. (10.1137/20M1357524)
  DOI : 10.1137/20M1357524
• The Morozov's principle applied to data assimilation problems
  
  • Bourgeois Laurent
  • Dardé Jérémi
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2022. This paper is focused on the Morozov's principle applied to an abstract data assimilation framework, with particular attention to three simple examples: the data assimilation problem for the Laplace equation, the Cauchy problem for the Laplace equation and the data assimilation problem for the heat equation. Those ill-posed problems are regularized with the help of a mixed type formulation which is proved to be equivalent to a Tikhonov regularization applied to a well-chosen operator. The main issue is that such operator may not have a dense range, which makes it necessary to extend well-known results related to the Morozov's choice of the regularization parameter to that unusual situation. The solution which satisfies the Morozov's principle is computed with the help of the duality in optimization, possibly by forcing the solution to satisfy given a priori constraints. Some numerical results in two dimensions are proposed in the case of the data assimilation problem for the Laplace equation. (10.1051/m2an/2022061)
  DOI : 10.1051/m2an/2022061
• On the approximation of electromagnetic fields by edge finite elements. Part 4: analysis of the model with one sign-changing coefficient
  
  • Ciarlet Patrick
  Numerische Mathematik, Springer Verlag, 2022, 152, pp.223-257. In electromagnetism, in the presence of a negative material surrounded by a classical material, the electric permittivity, and possibly the magnetic permeability, can exhibit a sign-change at the interface. In this setting, the study of electromagnetic phenomena is a challenging topic. We focus on the time-harmonic Maxwell equations in a bounded set $\Omega$ of ${\mathbb R}^3$, and more precisely on the numerical approximation of the electromagnetic fields by edge finite elements. Special attention is paid to low-regularity solutions, in terms of the Sobolev scale $({\boldsymbol{H}}^{\mathtt{s}}(\Omega))_{\mathtt{s}>0}$. With the help of T-coercivity, we address the case of one sign-changing coefficient, both for the model itself, and for its discrete version. Optimal a priori error estimates are derived. (10.1007/s00211-022-01315-x)
  DOI : 10.1007/s00211-022-01315-x