Publications

2009

• Numerical analysis of the generalized Maxwell equations (with an elliptic correction) for charged particle simulations
  
  • Ciarlet Patrick
  • Labrunie Simon
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2009, 19 (11), pp.1959-1994. When computing numerical solutions to the Vlasov--Maxwell equations, the source terms in Maxwell's equations usually fail to satisfy the continuity equation. Since this condition is required for the well-posedness of Maxwell's equations, it is necessary to introduce generalized Maxwell's equations which remain well-posed when there are errors in the sources. These approaches, which involve a hyperbolic, a parabolic and an elliptic correction, have been recently analyzed mathematically. The goal of this paper is to carry out the numerical analysis for several variants of Maxwell's equations with an elliptic correction. (10.1142/S0218202509004017)
  DOI : 10.1142/S0218202509004017
• Exact boundary conditions for time-harmonic wave propagation in locally perturbed periodic media
  
  • Fliss Sonia
  • Joly Patrick
  Applied Numerical Mathematics: an IMACS journal, Elsevier, 2009, 59 (9), pp.2155-2178. We consider the solution of the Helmholtz equation with absorption − u(x)−n(x)2(ω2 + ıε)u(x) = f (x), x = (x, y), in a 2D periodic medium Ω = R2. We assume that f (x) is supported in a bounded domain Ωi and that n(x) is periodic in the two directions in Ωe = Ω \ Ωi . We show how to obtain exact boundary conditions on the boundary of Ωi ,ΣS that will enable us to find the solution on Ωi . Then the solution can be extended in Ω in a straightforward manner from the values on ΣS . The particular case of medium with symmetries is exposed. The exact boundary conditions are found by solving a family of waveguide problems. © 2008 IMACS. (10.1016/j.apnum.2008.12.013)
  DOI : 10.1016/j.apnum.2008.12.013