Mixed and discontinuous finite volume element schemes for the optimal control of immiscible flow in porous media

In this article we introduce a family of discretisations for the numerical approximation of optimal control problems governed by the equations of immiscible displacement in porous media. The proposed schemes are based on mixed and discontinuous finite volume element methods in combination with the optimise-then-discretise approach for the approximation of the optimal control problem, leading to nonsymmetric algebraic systems, and employing minimum regularity requirements. Estimates for the error (between a local reference solution of the infinite dimensional optimal control problem and its hybrid mixed/discontinuous approximation) measured in suitable norms are derived, showing optimal orders of convergence.