  In this paper, we propose a method for the construction of locally
conservative flux fields through a variation of the Generalized Multiscale
Finite Element Method (GMsFEM). The flux values are obtained through the use of
a Ritz formulation in which we augment the resulting linear system of the
continuous Galerkin (CG) formulation in the higher-order GMsFEM approximation
space. In particular, we impose the finite volume-based restrictions through
incorporating a scalar Lagrange multiplier for each mass conservation
constraint. This approach can be equivalently viewed as a constraint
minimization problem where we minimize the energy functional of the equation
restricted to the subspace of functions that satisfy the desired conservation
properties. To test the performance of the method we consider equations with
heterogeneous permeability coefficients that have high-variation and
discontinuities, and couple the resulting fluxes to a two-phase flow model. The
increase in accuracy associated with the computation of the GMsFEM pressure
solutions is inherited by the flux fields and saturation solutions, and is
closely correlated to the size of the reduced-order systems. In particular, the
addition of more basis functions to the enriched multiscale space produces
solutions that more accurately capture the behavior of the fine scale model. A
variety of numerical examples are offered to validate the performance of the
method.
