Brezzi-Pitkaranta stabilization and a priori error analysis for theStokes control
In this study, we consider a Brezzi-Pitkaranta stabilization scheme for the optimal control problem governed by stationary Stokes equation, using a P1- P1 interpolation for velocity and pressure. We express the stabilization as extra terms added to the discrete variational form of the problem. We first prove the stability of the finite element discretization of the problem. Then, we derive a priori error bounds for each variable and present a numerical example to show the effectiveness of the stabilization clearly.
[1] Abergel, F., Temam, R., On some optimal control problems in fluid mechanics, Theoret. Comput. Fluid Mech., 1 (6), 303-325, 1990.
[2] Arnold, D., Brezzi, F., Fortin, M., A stable finite element for the Stokes equations, Calcolo 21, 337- 344, 1984.
[3] Adams, R.A., Sobolev Spaces, Academic Press, New York, 1975.
[4] Baiocchi, C., Brezzi, F. and Franca, L., Virtual bubbles and Galerkin-least-squares type methods (Ga. L. S.), Comput. Methods Appl. Mech. Engrg., 105, 125-141, 1993.
[5] Becker, R. and Hansbo, P., A simple pressure stabilization method for the Stokes equation, Commun. Numer. Meth. Engng, 24, 2008.
[6] Barrenechea, G.R. and Blasco, J., Pressure stabilization of finite element approximations of timedependent incompressible flow problems, Computer Methods in Applied Mechanics and Engineering, 197:1-4, 219-231, 2007.
[7] Bochev, P. and Gunzburger, M., Least-squares finite-element methods for optimization and control for the Stokes equations, Comput. Math. Appl., 48, 1035-1057, 2004.
[8] Brezzi, F. and Pitkaranta, J., On the stabilization of finite element approximations of the Stokes problem, in Efficient Solution of Elliptic Systems, W. Hackbush, ed., vol. 10 of Notes on Numerical Fluid Mechanics, Vieweg, 11-19, 1984.
[9] Brooks, A. and Hughes, T., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 32, 199-259, 1982.
[10] Casas, E., Optimality conditions for some control problems of turbulent flows, in: Flow Control, Minneapolis, MN, 1992, in: IMA Vol. Math. Appl., vol. 68, Springer, New York, 127-147, 1995.
[11] Codina, R. and Blasco, J., Analysis of a pressurestabilized finite element approximation of the stationary Navier-Stokes equations, Numer. Math., 87, 59-81, 2000.
[12] Douglas, J. and Wang, J., An absolutely stabi-lized finite element method for the Stokes prob-lem, Math. Comp., 52,495-508, 1989.
[13] Franca, L. and Hughes, T., Convergence analy-ses of Galerkin-Least-Squares methods for sym-metric advective-diffusive forms of the Stokes andincompressible Navier-Stokes equations, Comput.Methods. Appl. Mech. Engrg., 105 ,285-298.,1993.
[14] De los Reyes, J.C., Meyer, C. and Vexler, B., Fi-nite element error analysis for state-constrainedoptimal control of the Stokes equations, WIASPreprint No. 1292, 2008
[15] Gunzburger, M., Finite Element Methods for Vis-cous Incompressible Flows: A Guide to The-ory, Practice and Algorithms, Academic Press,Boston, 1989.
[16] Hecht, N., New development in FreeFem++, J.Numer. Math. 20, no. 3-4, 251-265, 2012.
[17] Heinkenschloss, M. and Leykekhman, D., Lo-cal Error Estimates for SUPG Solutions ofAdvection-Dominated Elliptic Linear-QuadraticOptimal Control Problems, Rice University, Hous-ton, CAAM Technical Report TR08-30, 2008.
[18] Hinze, M., Yan, N. and Zhou, Z., Variational dis-cretization for optimal control governed by con-vection dominated diffusion equations, J. Com-put. Math., 27, 237-253, 2009.
[19] Latche, J.-C. and Vola, D., Analysis of the Brezzi-Pitkaranta Stabilized Galerkin Scheme for Creep-ing Flows of Bingham Fluids, Siam J. Numer.Anal., 42:3, 1208-1225, 2004.
[20] Layton, W.J., A connection between subgrid scale eddy viscosity and mixed methods, Appl. Math. Comput., 133, 147-157, 2002.
[21] Lions, J.L., Optimal control of systems governed by partial differential equations, Springer Verlag, New York, 1971.
[22] Rees, T. and Wathen, A.J., Preconditioning Iterative Methods for the Optimal Control of the Stokes Equations, SIAM J. Sci. Comput., 33(5), 2903-2926, 2011.
[23] R¨osch, A. and Vexler, B., Optimal Control of the Stokes Equations: A Priori Error Analysis for Finite Element Discretization with Postprocessing, SIAM J. Numer. Anal., 44, 1903-1920, 2006.
[24] Stoll, M. and Wathen, A., All-at-once solution of time-dependent Stokes control, Journal of Computational Physics, 231(1), 498-515, 2013.
[25] Sun, T., Discontinuous Galerkin finite element method with interior penalties for convection diffusion optimal control problem. Int. J. Numer. Anal. Model. 7, 87-107, 2010.
[26] Tr¨oltzsch, F., Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Amer Mathematical Society, 2010.
[27] Zhou, Z. and Yan, N., The local discontinuous Galerkin method for optimal control problem governed by convection-diffusion equations, International Journal of Numerical Analysis & Modeling 7, 681-699, 2010.