Abstract:
Various approaches to extend the finite element methods to non-traditional elements (pyramids, prisms and polyhedra) have been developed over last decade. Building of basis functions for such elements is a challenging task and may require extensive geometry analysis. I present a novel mimetic discretization methodology that uses only surface representation of discrete unknowns to build stiffness and mass matrices. Since no extension inside a mesh element is required, practical implementation of the mimetic methods becomes simple for non-convex and degenerate elements.
The mimetic discretization methods have been developed originally
using a finite difference framework. However, their strong
connection with finite elements allows us to present them as a
generalization of finite element methods to polygonal and polyhedral
meshes. We present the mimetic discretization of the Stokes problem.
We start with the mimetic generalization of the unstable 
finite element. First, we prove that it is stable on a large
range of polygonal meshes. Second, we show how to stabilize the
remaining cases by adding a small number of bubble functions to
selected mesh edges. A simple strategy for selecting such edges
will be proposed and verified with numerical experiments.
This seminar is easily accessible to persons with disabilities. For more information or for assistance, please contact the Mathematics Department at 743-3500.