DISCONTINUOUS FINITE VOLUME METHODS FOR OPTIMAL CONTROL OF BRINKMAN EQUATIONS

Abstract:   In this talk, I will discuss a discontinuous finite volume (DFV) approximation of distributed optimal control problems governed Brinkman equations (that describe flow of an incompressible viscous fluid through a porous medium) written in terms of velocity and pressure.  Before discussing optimal control of Brinkman equations, I will present briefly on DFV approximations of optimal control problems governed by semilinear elliptic, parabolic and hyperbolic equations. The discretization of state and costate velocity and pressure fields follows a lowest order discontinuous finite volume scheme, whereas three different strategies are used for control approximation: variational discretization approach (in which control set is implicitly discretized by a projection of discrete costate variables), as well as piecewise constant and piecewise linear discretizations. As the resulting discrete optimal system is non-symmetric, the optimize-then-discretize approach is employed to approximate the control problem. A priori error estimates for velocity, pressure and control in suitable norms is derived and numerical examples are presented to illustrate the performance of the proposed method and to confirm the predicted accuracy under various scenarios including 2D and 3D cases.