| Speaker: | Pavel Bochev, Applied mathematics and applications department, Sandia National Laboratories |
| Title: | Rehabilitation of the lowest-order Raviart-Thomas element on quadrilateral grids |
| Time: | 3:00-4:00 pm |
| Place: | Blocker 628 |
A recent study by Arnold et. al. points out that the convergence of finite element methods that use H(div)-compatible finite element spaces deteriorates on non-affine quadrilateral grids. This deterioration is particularly troublesome for the lowest-order Raviart-Thomas elements, because it implies loss of convergence in some norms for finite element solutions of mixed and least-squares methods. In this talk we show that a reformulation of finite element methods in terms the so-called natural mimetic divergence operator  restores the order of convergence.
Reformulations of mixed Galerkin and least-squares methods for the Darcy equation illustrate our approach. We prove that reformulated methods converge optimally with respect to a norm involving the mimetic divergence operator. Furthermore, we prove that standard and reformulated versions of the mixed Galerkin method actually coincide, but that the two versions of the least-squares method are genuinely different. The surprising conclusion is that the degradation of convergence in the mixed method on non-affine quadrilateral grids is superficial, and that the lowest order Raviart-Thomas elements are safe to use in this method. However, the breakdown in the least-squares method is real, and there one should use our proposed reformulation.
This is joint work with D. Ridzal.
Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94-AL85000.
Last revised: 09/27/07 By: abnersg@math