Estimations d'erreur a priori de la méthode de Lagrange-Galerkin pour les systèmes de type Kazhikhov-Smagulov

Kazhikhov-Smagulov type systems are a subclass of non-homogeneous, incompressible Navier-Stokes equations where density is subject to diffusion, as in mixtures of gases of different densities. An algorithm is devised for these systems, the time discretization being based on a backward-Euler scheme together with the method of characteristics, and a mixed density-velocity-pressure (Pk,Pk,Pk−1) finite element method is used for the space discretization in Rd, d=2,3. Under the constraint that k>d−1 and Δt=Chr, with r∈]d,2k+2−d[, we give optimal error bounds O(Δt+hk) for the time step Δt and the mesh size h. To cite this article: J. Étienne, P. Saramito, C. R. Acad. Sci. Paris, Ser. I 341 (2005).