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
• On the Half-Space Matching Method for Real Wavenumber
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chandler-Wilde Simon N
  • Fliss Sonia
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2022, 82 (4). The Half-Space Matching (HSM) method has recently been developed as a new method for the solution of 2D scattering problems with complex backgrounds, providing an alternative to Perfectly Matched Layers (PML) or other artificial boundary conditions. Based on half-plane representations for the solution, the scattering problem is rewritten as a system coupling (1) a standard finite element discretisation localised around the scatterer and (2) integral equations whose unknowns are traces of the solution on the boundaries of a finite number of overlapping half-planes contained in the domain. While satisfactory numerical results have been obtained for real wavenumbers, well-posedness and equivalence of this HSM formulation to the original scattering problem have been established only for complex wavenumbers. In the present paper we show, in the case of a homogeneous background, that the HSM formulation is equivalent to the original scattering problem also for real wavenumbers, and so is well-posed, provided the traces satisfy radiation conditions at infinity analogous to the standard Sommerfeld radiation condition. As a key component of our argument we show that, if the trace on the boundary of a half-plane satisfies our new radiation condition, then the corresponding solution to the half-plane Dirichlet problem satisfies the Sommerfeld radiation condition in a slightly smaller half-plane. We expect that this last result will be of independent interest, in particular in studies of rough surface scattering. (10.1137/21M1459216)
  DOI : 10.1137/21M1459216
• The Complex-Scaled Half-Space Matching Method
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chandler-Wilde Simon N.
  • Fliss Sonia
  • Hazard Christophe
  • Perfekt Karl-Mikael
  • Tjandrawidjaja Yohanes
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2022, 54 (1), pp.512-557. The Half-Space Matching (HSM) method has recently been developed as a new method for the solution of 2D scattering problems with complex backgrounds, providing an alternative to Perfectly Matched Layers (PML) or other artificial boundary conditions. Based on half-plane representations for the solution, the scattering problem is rewritten as a system of integral equations in which the unknowns are restrictions of the solution to the boundaries of a finite number of overlapping half-planes contained in the domain: this integral equation system is coupled to a standard finite element discretisation localised around the scatterer. While satisfactory numerical results have been obtained for real wavenumbers, wellposedness and equivalence to the original scattering problem have been established only for complex wavenumbers. In the present paper, by combining the HSM framework with a complex-scaling technique, we provide a new formulation for real wavenumbers which is provably well-posed and has the attraction for computation that the complex-scaled solutions of the integral equation system decay exponentially at infinity. The analysis requires the study of double-layer potential integral operators on intersecting infinite lines, and their analytic continuations. The effectiveness of the method is validated by preliminary numerical results. (10.1137/20M1387122)
  DOI : 10.1137/20M1387122
• 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