Title: Second-order accurate discretization method for diffusion problems with tensor coefficients on polyhedral meshes.

Speaker: K. Lipnikov, LANL

Date/Time: Monday, August 20, 2007, 10:30 - 11:30 am

Location: CSRI Building/Room 279 (NM)

Brief Abstract: A mimetic finite difference (MFD) method preserves essential properties of the continuum differential operators, such as conservation laws, solution symmetries, and the fundamental identities and theorems of vector and tensor calculus. For the linear diffusion problem, the MFD method mimics the Gauss divergence theorem to enforce the local mass conservation, the symmetry between the continuous gradient and divergence operators to guarantee symmetry and positivity of the resulting discrete operator, and the null spaces of the involved operators to have stability of the discretization.

In the talk, I present the MFD method for solving the diffusion problem with tensor coefficients on unstructured polyhedral meshes consisting of arbitrary elements: tetrahedrons, pyramids, hexahedrons, degenerated and non-convex polyhedrons, etc. The MFD method uses simple geometric characteristics like normals and areas of mesh faces; therefore, it can be applied, for example, to AMR meshes with hanging nodes in exactly the same manner as to tetrahedral meshes.

In some sense, the MFD method lies between the standard mixed finite element (MFE) and finite volume (FV) methods. In the FV method, the fluxes are defined only at interfaces between mesh elements and a finite difference formula is used to discretize the constitutive equation. In the MFE method, a polynomial representation of the vector field inside each mesh element is used to define the inner product between vectors and then to write the constitutive equation by duality. This, however, can be done only for simple geometrical shapes. In the MFD method, there is notion of the inner product between vectors but the vector field inside a mesh element is never introduced explicitly. It is like a "guardian angel" who helps us prove convergence results but cannot be seen in the method formulation. Since the inner product is derived without any reconstruction, the practical implementation of the method is quite simple.

It is known that the classical MFE method with the lowest order Raviart-Thomas elements does not converge on a sequence of randomly perturbed hexahedral meshes. We shall show that the MFD method produces a second-order accurate discretization scheme on such meshes.

This is a joint work with Franco Brezzi, Mikhail Shashkov, and Valeria Simonchini.

CSRI POC: Pavel Bochev , (505) 844-1990