A Finite Element Approximation of Grazing Collisions

In this article, we propose a finite element discretization of the Boltzmann-Lorentz operator for which it is possible to define a grazing collision limit. We illustrate this discretization by considering the evolution of a system of particles in a slab, subject to collisions both with the boundaries and with themselves. A comparison is made with isotropic collisions or with the Laplace-Beltrami operator. Moreover, in the case of multiplying boundary conditions, it is proven by numerical simulations the existence of a critical value for the absorption coefficient, which is independent on the grazing collision parameter. Finally, the focalization of a beam is studied and the numerical results are compared with previous simulations. AMS Classification: 35B40; 65N30; 82C40. *Correspondence: B. Lucquin-Desreux, Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Boite Courrier 187, 75252 Paris, Cedex 05, France; E-mail: [email protected]. 120024765_TT32_03-04_R1_071703