Local flux mimetic finite difference method for diffusion problems

ABSTRACT. A successful discretization method inherits or mimics 
fundamental properties of the PDE such as conservation laws, 
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 for solving 
problems with full tensor coefficients on unstructured polygonal
and polyhedral meshes. These meshes may include arbitrary elements: 
tetrahedrons, pyramids, hexahedrons, degenerated and non-convex 
polyhedrons and generalized polyhedrons.

The MFD method for diffusion problems leads usually to a symmetric 
finite difference scheme. However, the resulting algebraic system is 
of a saddle-point type and couples the velocity (vector variable) and 
the pressure (scalar variable) unknowns. Elimination of the velocity 
unknowns results in a cell-centered discretization scheme with a 
non-local stencil. In this talk, I present a MFD method that can be 
reduced to a cell-centered scheme with a local stencil. Under certain 
assumptions, first-order convergence is proved for both variables and 
second-order convergence is proved for the pressure. Numerical results 
confirm the theory.