Event Type:

Department Colloquium

Date/Time:

Friday, March 14, 2008 - 09:00

Location:

Kidd 364

Guest Speaker:

Institution:

Los Alamos National Laboratory

Abstract:

A successful discretization method inherits or mimics fundamental properties of PDEs such as topology, conservation, symmetries, positivity structures and maximum principles. Construction of such a method is made more difficult when the mesh is distorted so that it can conform and adapt to the physical domain and problem solution. The talk is about one such method - the mimetic finite difference (MFD) method. The MFD method can be applied to solve PDEs with full tensor coefficients on unstructured polygonal and polyhedral meshes. These meshes may include arbitrary elements: tetrahedrons, pyramids, hexahedrons, degenerated and non-convex polyhedrons, generalized polyhedrons, etc. I present a general framework of the MFD method, give examples of discrete gradient, divergence and curl operators on polygonal and polyhedral meshes, and review existing theoretical results including convergence estimates, orthogonal decomposition theorem, etc. I'll show how the MFD framework can be used to derive and analyze new multi-point flux approximation methods. The MFD method has been applied successfully to several applications including diffusion, electromagnetics, acoustics, and gasdynamics. It was proved that for diffusion problem, the MFD method produces a family of schemes with equivalent properties. For simplicial meshes, this family contains the classical mixed finite element methods with the lowest order Raviart-Thomas elements. This talk is complementary to a more specialized talk on the MFD method for diffusion problems given the same day at the Applied Mathematics Seminar.