A high-order mimetic method on unstructured polyhedral meshes for the diffusion equation

• We present a new family of mimetic finite difference schemes for diffusion problems.
• The formulation of the method works for 3-D unstructured polyhedral meshes.
• The degrees of freedom are the solution moments on mesh faces and inside mesh cells.
• Higher order schemes are built using higher order moments.
• The method is built by imposing a stability and consistency condition.

We present a new family of mimetic finite difference schemes for solving elliptic partial differential equations in the primal form on unstructured polyhedral meshes. These mimetic discretizations are built to satisfy local consistency and stability conditions. The consistency condition is an exactness property, i.e., the mimetic schemes are exact when the solution is a polynomial of an assigned degree. The stability condition ensures the well-posedness of the method. The degrees of freedom are the solution moments on mesh faces and inside mesh cells. Higher order schemes are built using higher order moments. The developed schemes are verified numerically on diffusion problems with constant and spatially variable (possibly, discontinuous) tensorial coefficients.