Grad‐Div Stabilization for Approximating Control of Thermally Convective Flows With Mixed Finite Elements

ABSTRACT We investigate approximating control of viscous incompressible thermally convective flows using inf‐sup stable mixed finite elements with grad‐div stabilization. The proposed method augments the Galerkin finite element method with a grad‐div stabilization term enhancing mass conservation and improving the performance of the algorithm. We analyse a mixed finite element method‐based semi‐discretization of the grad‐div stabilized optimality system from which optimal control can be computed. An optimal order error estimate of the approximations of the optimality system is derived in and ‐norms without nonlocal compatibility conditions. Numerical results are presented by solving a control problem that involves minimizing vorticity in a mixed convective flow past a backward‐facing step channel. Numerical experiments show the feasibility and applicability of the grad‐div stabilization for approximating control of thermally convective flows with mixed finite elements.