A priori error estimates for discontinuous finite volume discretizations of the Brinkman optimal control problem

In this paper we discuss a discontinuous finite volume method for the approximation of distributed optimal control problem governed by the Brinkman equations written in terms of velocity and pressure. An additional force field is sought such that it produces a velocity matching a desired, known value. The discretization of state and co-state velocity and pressure fields follows a lowest order discontinuous finite volume scheme, whereas three different approaches are used for the control approximation: variational discretization, element-wise constant, and element-wise linear functions. We employ the optimize-then-discretize approach to approximate the control problem, and the resulting discretized formulation is non-symmetric. We derive a priori error estimates for velocity, pressure, and control in natural norms. A set of numerical examples is finally presented to illustrate the performance of the method and to confirm the predicted accuracy of the state, co-state and control approximations under various scenarios including 2D and 3D cases.