Publications

2009

• Fast and accurate computation of layer heat potentials
  
  • Li Jing-Rebecca
  • Greengard Leslie
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2009. We discuss the numerical evaluation of single and double layer heat potentials in two dimensions on stationary and moving boundaries. One of the principal difficulties in designing high order methods concerns the local behavior of the heat kernel, which is both weakly singular in time and rapidly decaying in space. We show that standard quadrature schemes suffer from a poorly recognized form of inaccuracy, which we refer to as geometrically-induced stiffness, but that rules based on product integration of the full heat kernel in time are robust. When combined with previously developed fast algorithms for the evolution of the history part of layer potentials, diffusion processes in complex, moving geometries can be computed accurately and in nearly optimal time.
• 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
• Numerical resolution of the wave equation on a network of slots
  
  • Semin Adrien
  , 2009, pp.35. In this technical report, we present a theoretical and numerical model to simulate wave propagation in finite networks of rods with both classical Kirchhoff conditions and Improved Kirchhoff conditions at the nodes of the networks. One starts with the continuous framework, then we discretize the problem using finite elements with the mass lumping technic introduced by G.~Cohen and P.~Joly. Finally, we show an implementation of the obtained numeric scheme in a homemade code written in C++ in collaboration with K.~Boxberger, some results and some error estimates.
• Space-time mesh refinement for discontunuous Galerkin methods for symmetric hyperbolic systems
  
  • Ezziani Abdelaâziz
  • Joly Patrick
  Journal of Computational and Applied Mathematics, Elsevier, 2009, 234 (6), pp.1886-1895. We present a new non-conforming space-time mesh refinement method for the symmetric first order hyperbolic system. This method is based on the one hand on the use of a conservative higher order discontinuous Galerkin approximation for space discretization and a finite difference scheme in time, on the other hand on appropriate discrete transmission conditions between the grids. We use a discrete energy technique to drive the construction of the matching procedure between the grids and guarantee the stability of the method. (10.1016/j.cam.2009.08.094)
  DOI : 10.1016/j.cam.2009.08.094