  This article presents a high order conservative flux optimization (CFO)
finite element method for the elliptic diffusion equations. The numerical
scheme is based on the classical Galerkin finite element method enhanced by a
flux approximation on the boundary of a prescribed set of arbitrary control
volumes (either the finite element partition itself or dual voronoi mesh, etc).
The numerical approximations can be characterized as the solution of a
constrained-minimization problem with constraints given by the flux
conservation equations on each control volume. The discrete linear system is a
typical saddle-point problem, but with less number of degrees of freedom than
the standard mixed finite element method, particularly for elements of high
order. Moreover, the numerical solution of the proposed scheme is of
super-closeness with the finite element solution. Error estimates of optimal
order are established for the numerical flux as well as the primary variable
approximations. We present several numerical studies in order to verify
convergence of the CFO schemes. A simplified two-phase flow in highly
heterogeneous porous media model problem will also be presented. The numerical
results show obvious advantages of applying high order CFO schemes.
