Symmetric error estimates for discontinuous Galerkin time-stepping schemes for optimal control problems constrained to evolutionary Stokes equations. (English) Zbl 1328.65203
Summary: We consider fully discrete finite element approximations of a distributed optimal control problem, constrained by the evolutionary Stokes equations. Conforming finite element methods for spatial discretization combined with discontinuous time-stepping Galerkin schemes are being used for the space-time discretization. Error estimates are proved under weak regularity hypotheses for the state, adjoint and control variables. The estimates are also applicable when high order schemes are being used. Computational examples validating our expected rates of convergence are also provided.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 49J20 | Existence theories for optimal control problems involving partial differential equations |
Keywords:
discontinuous time-stepping schemes; finite element approximations; Stokes equations; velocity tracking problem; distributed controls; error estimatesSoftware:
FreeFem++References:
| [1] | Abergel, F., Temam, R.: On some control problems in fluid mechanics. Theor. Comput. Fluid Dyn. 1, 303-326 (1990) · Zbl 0708.76106 · doi:10.1007/BF00271794 |
| [2] | Apel, T., Flaig, T.: Crank-Nicolson schemes for optimal control problems with evolution equations. SIAM J. Numer. Anal. 50, 1484-1512 (2012) · Zbl 1248.49041 · doi:10.1137/100819333 |
| [3] | Casas, E.; Sritharan, SS (ed.), An optimal control problem governed by the evolution Navier-Stokes equations (1998), Philadelphia |
| [4] | Casas, E., Chrysafinos, K.: A discontinuous Galerkin time stepping scheme for the velocity tracking problem. SIAM. J. Numer. Anal. 50, 2281-2306 (2012) · Zbl 1273.49035 · doi:10.1137/110829404 |
| [5] | Casas, E., Chrysafinos, K.: Error estimates for the discretization of the velocity tracking problem. http://www.math.ntua.gr/ chrysafinos. Accessed 1 Sept 2014 · Zbl 1426.76232 |
| [6] | Chrysafinos, K.: Discontinuous Galerkin finite element approximations for distributed optimal control problems constrained to parabolic PDE’s. Int. J. Numer. Anal. Model. 4, 690-712 (2007) · Zbl 1149.65046 |
| [7] | Chrysafinos, K.: Analysis and finite element approximations for distributed optimal control problems for implicit parabolic equations. J. Comput. Appl. Math. 231, 327-348 (2009) · Zbl 1179.65076 · doi:10.1016/j.cam.2009.02.092 |
| [8] | Chrysafinos, K., Karatzas, E.: Symmetric error estimates for discontinuous Galerkin approximations for an optimal control problem associated to semilinear parabolic PDEs. Discret. Contin. Dynam. Syst. B 17, 1473-1506 (2012) · Zbl 1247.65080 · doi:10.3934/dcdsb.2012.17.1473 |
| [9] | Chrysafinos, K., karatzas, E.: Error estimates for discontinuous Galerkin time-stepping schemes for Robin boundary control problems constrained to parabolic pdes’, http://www.math.ntua.gr/ chrysafinos. Accessed 1 Sept 2014 · Zbl 1355.65082 |
| [10] | Chrysafinos, K., Walkington, N.J.: Discontinuous Galerkin approximations of the Stokes and Navier-Stokes equations. Math. Comput. 79, 2135-2167 (2010) · Zbl 1273.76077 · doi:10.1090/S0025-5718-10-02348-3 |
| [11] | Ciarlet, P.: The Finite Element Method for Elliptic Problems. SIAM Classics, Philadelphia (2002) · Zbl 0999.65129 · doi:10.1137/1.9780898719208 |
| [12] | Constantin, P.: Navier-Stokes equations. The University of Chicago press, Chicago (1988) · Zbl 0687.35071 |
| [13] | Dautray, R., Lions, J.-L.: Mathematical analysis and Numerical Methods for Science and Technology, vol. 6. Spinger-Verlag, Berlin (1993) · Zbl 0802.35001 |
| [14] | Deckelnick, K., Hinze, M.: Error estimates in space and time for tracking type control of the instationary Stokes system. Int. Ser. Numer. Math. 143, 87-103 (2002) · Zbl 1059.93053 |
| [15] | Deckelnick, K., Hinze, M.: Semidiscretization and error estimates for distributed control of the instationary Navier-Stokes equations. Numer. Math. 97, 297-320 (2004) · Zbl 1055.76027 · doi:10.1007/s00211-003-0507-4 |
| [16] | de los Reyes, J.-C., Kunisch, K.: A semi-smooth Newton method for control constrained boundary control of the Navie-Stokes equtions. Nonlinear Anal. 62, 1289-1316 (2005) · Zbl 1080.49024 · doi:10.1016/j.na.2005.04.035 |
| [17] | Evans, L.: Partial Differential Equations. AMS, Providence (1998) · Zbl 0902.35002 · doi:10.1090/gsm/019 |
| [18] | Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes. Springer-Verlag, New York (1986) · Zbl 0585.65077 · doi:10.1007/978-3-642-61623-5 |
| [19] | Gunzburger, MD, Perspectives in flow control and optimization (2003), Philadelphia · Zbl 1088.93001 |
| [20] | Gunzburger, M.D., Manservisi, S.: Analysis and approximation of the velocity tracking problem for Navier-Stokes flows with distributed control. SIAM J. Numer. Anal. 37, 1481-1512 (2000) · Zbl 0963.35150 · doi:10.1137/S0036142997329414 |
| [21] | Gunzburger, M.D., Manservisi, S.: The velocity tracking problem for Navier-Stokes flows with bounded distributed control. SIAM J. Control Optim. 37, 1913-1945 (2000) · Zbl 0938.35118 · doi:10.1137/S0363012998337400 |
| [22] | Gunzburger, M.D., Manservisi, S.: The velocity tracking problem for Navier-Stokes flow with boundary control. SIAM J. Control Optim. 39, 594-634 (2000) · Zbl 0991.49002 · doi:10.1137/S0363012999353771 |
| [23] | Hecht, F.: FreeFem++, 3rd edn, Version 3.13, 2011. http://www.freefem.org/ff++. Accessed 1 Aug 2014 · Zbl 1266.68090 |
| [24] | Heywood, J., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second order estimates for spatial discretization. SIAM J. Numer. Anal. 19, 275-311 (1982) · Zbl 0487.76035 · doi:10.1137/0719018 |
| [25] | Hinze, M.: A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 30, 45-61 (2005) · Zbl 1074.65069 · doi:10.1007/s10589-005-4559-5 |
| [26] | Hinze, M., Kunisch, K.: Second order methods for optimal control of time-dependent fluid flow. SIAM J. Control Optim. 40, 925-946 (2001) · Zbl 1012.49026 · doi:10.1137/S0363012999361810 |
| [27] | Hinze, M., Kunisch, K.: Second order methods for boundary control of the instationary Navier-Stokes system. ZAMMZ. Angew. Math. Mech. 84, 171-187 (2004) · Zbl 1042.35047 · doi:10.1002/zamm.200310094 |
| [28] | Hou, L.S.: Error estimates for semidiscrete finite element approximation of the evolutionary Stokes equations under minimal regularity assumptions. J. Sci. Comput. 16, 287-317 (2001) · Zbl 0996.76048 · doi:10.1023/A:1012869611793 |
| [29] | Kunisch, K., Vexler, B.: Constrained Dirichlet boundary control in \[L^2\] L2 for a class of evolution equations. SIAM J. Control Optim. 46(5), 1726-1753 (2007) · Zbl 1144.49003 · doi:10.1137/060670110 |
| [30] | Lasiecka, I., Triggiani, R.: Control Theory for Partial Differential Equations. Cambridge University Press, Cambridge (2000) · Zbl 0942.93001 · doi:10.1017/CBO9781107340848 |
| [31] | Lions, J.-L.: Some Aspects of the Control of Distributed Parameter Systems. SIAM Publications, Philadelphia (1972) · Zbl 0275.49001 · doi:10.1137/1.9781611970616 |
| [32] | Liu, W.-B., Ma, H.-P., Tang, T., Yan, N.: A posteriori error estimates for DG time-stepping method for optimal control problems governed by parabolic equations. SIAM J. Numer. Anal. 42(3), 1032-1061 (2004) · Zbl 1085.65054 · doi:10.1137/S0036142902397090 |
| [33] | Meidner, D., Vexler, B.: Adaptive space-time finite element methods for parabolic optimization problems. SIAM J. Control Optim. 46, 116-142 (2007) · Zbl 1149.65051 · doi:10.1137/060648994 |
| [34] | Meidner, D., Vexler, B.: A priori error estimates for space-time finite element discretization of parabolic optimal control problems. Part I: problems without control constraints. SIAM J. Control Optim. 47, 1150-1177 (2008) · Zbl 1161.49026 · doi:10.1137/070694016 |
| [35] | Meidner, D., Vexler, B.: A priori error estimates for space-time finite element discretization of parabolic optimal control problems. Part II: problems with control constraints. SIAM J. Control Optim. 47, 1301-1329 (2008) · Zbl 1161.49035 · doi:10.1137/070694028 |
| [36] | Neittaanmaki, P., Tiba, D.: Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications. M. Dekker, New York (1994) · Zbl 0812.49001 |
| [37] | Neitzel, I., Vexler, B.: A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems. Numer. Math. 120, 345-386 (2012) · Zbl 1245.65074 · doi:10.1007/s00211-011-0409-9 |
| [38] | Richter, T., Springer, A., Vexler, B.: Efficient numerical realization of discontinuous Galerkin methods for temporal discretization of parabolic problems. Numer. Math. 124, 151-182 (2013) · Zbl 1268.65138 · doi:10.1007/s00211-012-0511-7 |
| [39] | Springer, A., Vexler, B.: Third order convergent time discretization for parabolic optimal control problems with control constraints, Comput. Optim. Appl. 57(1), 205-240 (2014) · Zbl 1283.49033 |
| [40] | Sritharan, S.S.: Optimal Control of Viscous Flow. SIAM, Philadelphia (1998) · Zbl 0920.76004 · doi:10.1137/1.9781611971415 |
| [41] | Temam, R.: Navier-Stokes Equations. American Mathematical Society, Chelsea, Providence (2001) · Zbl 0981.35001 |
| [42] | Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Spinger-Verlag, Berlin (1997) · Zbl 0884.65097 · doi:10.1007/978-3-662-03359-3 |
| [43] | Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Graduate Studies in Mathematics. AMS, Providence (2010) · Zbl 1195.49001 · doi:10.1090/gsm/112 |
| [44] | Tröltzsch, F., Wachsmuth, D.: Second-order suficcient optimality conditions for the optimal control of Navier-Stokes equations. ESAIM COCV 12, 93-119 (2006) · Zbl 1111.49017 · doi:10.1051/cocv:2005029 |
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