  A solution technique is proposed for flows in porous media that guarantees
local conservation of mass. We first compute a flux field to balance the mass
source and then exploit exact co-chain complexes to generate a solenoidal
correction. A reduced basis method based on proper orthogonal decomposition is
employed to construct the correction and we show that mass balance is ensured
regardless of the quality of the reduced basis approximation. The method is
directly applicable to mixed finite and virtual element methods, among other
structure-preserving discretization techniques, and we present the extension to
Darcy flow in fractured porous media.
