| Speaker: | Vidar Thomee, |
| Title: | On Maximum-Principles in Elliptic and Parabolic Finite Element Problems |
| Time: | 3:00 pm - 4:00pm |
| Place: | Blocker 628 |
We consider finite element discretizations of Dirichlet's problem for Laplace's equation and the corresponding initial-boundary value problem for the homogeneous heat equation. We survey work on discrete versions of the associated maximum-principles, using piecewise linear finite elements on triangulations of the underlying spatial domain. In the elliptic case the maximum-principle holds if the triangulation is of Delaunay type, but that this is is not a necessary condition. For the spatially semidiscrete parabolic problem, the maxiumum-principle does not hold in general, but, as was shown by Fujii in 1973, it is valid for the lumped mass variant when the triangulation is of Delaunay type. We show that theseconditions are essentially sharp. We also study conditions for the solution operator acting on the discrete initial data, with homogeneous lateral boundary conditions, to be a contraction in the maximum-norm, or a positive operator.
Last revised: 04/23/08 By: abnersg@math