Superconvergence in some locally conservative discretization methods

Ivan Yotov
University of Pittsburgh

Mixed finite element methods (MFEM), control-volume mixed finite
element methods (CVMFEM), and mimetic finite difference methods (MFDM)
are locally mass conservative discretization methods that perform well
on diffusion-type problems with rough grids and coefficients. All
methods provide accurate approximations of both the scalar variable
(pressure) and its flux (velocity). MFEM are naturally formulated as
variational methods. CVMFEM were originally developed as finite volume
methods, while MFDM are based on discrete operators that preserve
critical properties of the differential operators.

We will start with a brief overview of known techniques and results in
superconvergence of MFEM. We will then discuss how recent formulations
of CVMFEM and MFDM as variational methods can be employed to establish
superconvergence for these methods in both pressure and velocity.