| Speaker: | Prof. Vivette Girault, Laboratoire d'Analyse Numérique, Université Pierre et Marie Curie, Paris, France |
| Title: | Two-grid finite-element schemes in three dimensions for the steady and for the time-dependent Navier-Stokes equations |
| Time: | 3:00-4:00 pm |
| Place: | Blocker 503 |
The idea of a two-grid method for solving the steady Navier-Stokes equations consists in solving first the fully nonlinear discrete problem on a coarse grid (with mesh-size H) and next linearizing the discrete problem on a fine grid (with mesh-size h) by substituting the velocity computed on the coarse grid, say uH, into the nonlinear term. The motivation for this splitting is that the contribution of uH to the error is measured in a norm that is weaker than the regular H1 norm. In three dimensions, it is the L3 norm; and by a duality argument, the error ||uH-u||L3 is of the order of H3/2. Therefore, if h and H are related by h = H3/2 one can expect that the scheme after these two steps has the same order of convergence as if the fully nonlinear problem had been solved directly on the fine grid. This result is optimal in the sense that it applies to an arbitrary Lipschitz polyhedron. The idea is the same for a semi-discretization of the time-dependent Navier-Stokes equations, but the proofs require more regularity and do not seem to be optimal, as far as the assumptions on the solution and on the domain are concerned.
Last revised Apr 2, 2001. mail to Webmaster