MIXED DISCONTINUOUS GALERKIN TIME-STEPPING METHOD FOR LINEAR PARABOLIC OPTIMAL CONTROL PROBLEMS

Tianliang Hou1,2, Yanping Chen3

1. 1. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China;
2. School of Mathematics and Statistics, Beihua University, Jilin 132013, China;
3. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
• Supported by:

The work of T. Hou was supported by China Postdoctoral Science Foundation funded project (2013M542188). The work of Y. Chen was supported by National Science Foundation of China (91430104, 11271145), and Specialized Research Fund for the Doctoral Program of Higher Education (20114407110009).

Tianliang Hou, Yanping Chen. MIXED DISCONTINUOUS GALERKIN TIME-STEPPING METHOD FOR LINEAR PARABOLIC OPTIMAL CONTROL PROBLEMS[J]. Journal of Computational Mathematics, 2015, 33(2): 158-178.

In this paper, we discuss the mixed discontinuous Galerkin (DG) finite element approximation to linear parabolic optimal control problems. For the state variables and the co-state variables, the discontinuous finite element method is used for the time discretization and the Raviart-Thomas mixed finite element method is used for the space discretization. We do not discretize the space of admissible control but implicitly utilize the relation between co-state and control for the discretization of the control. We derive a priori error estimates for the lowest order mixed DG finite element approximation. Moveover, for the element of arbitrary order in space and time, we derive a posteriori L2(0, T;L2(Ω)) error estimates for the scalar functions, assuming that only the underlying mesh is static. Finally, we present an example to confirm the theoretical result on a priori error estimates.

CLC Number:

  N. Arada, E. Casas and F. Tröltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem, Comp. Optim. Appl., 23 (2002), 201-229. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer-Verlag., 95 (1991),65-187. Y. Chen, Superconvergence of quadratic optimal control problems by triangular mixed finite elements, Inter. J. Numer. Meths. Eng., 75 (2008), 881-898. Y. Chen, Y. Huang, W.B. Liu and N.N. Yan, Error estimates and superconvergence of mixed finite element methods for convex optimal control problems, J. Sci. Comput., 42 (2009), 382-403. Y. Chen and T. Hou, Superconvergence and L∞-error estimates of RT1 mixed methods for semilinear elliptic control problems with an integral constraint, Numer. Math. Theor. Meth. Appl. 5 (2012), 423-446. Y. Chen and T. Hou, Error estimates and superconvergence of RT0 mixed methods for a class of semilinear elliptic optimal control problems, Numer. Math. Theor. Meth. Appl., 6 (2013), 637-656. Y. Chen and W. B. Liu, A posteriori error estimates for mixed finite element solutions of convex optimal control problems, J. Comp. Appl. Math., 211 (2008), 76-89. Y. Chen, L. Liu and Z. Lu, A posteriori error estimates of mixed methods for parabolic optimal control problems, Numer. Funct. Anal. Optim., 31 (2010), 1135-1157. J. Douglas and J.E. Roberts, Global estimates for mixed finite element methods for second order elliptic equations, Math. Comp., 44 (1985), 39-52. K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem, SIAM J. Numer. Anal., 28 (1991), 43-77. K. Eriksson, C. Johnson and V. Thomée, Time discretization of parabolic problems by the discontinuous Galerkin method, RAIRO Modél. Math. Anal. Numér., 19 (1985), 611-643. W. Gong and N. Yan, A posteriori error estimate for boundary control problems governed by the parabolic partial differential equations, J. Comput. Math., 27 (2009), 68-88. J. Haslinger and P. Neittaanmäki, Finite Element Approximation for Optimal Shape Design, John Wiley and Sons, Chichester, UK, 1989. L. Hou and J.C. Turner, Analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls, Numer. Math., 71 (1995), 289-315. M. Hinze, A variational discretization concept in control constrained optimization: the linearquadratic case, Comp. Optim. Appl., 30 (2005), 45-63. M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with pde constraints. Mathematical Modelling: Theory and Applications, Springer, 2008. P. Houston and E. Süli, A posteriori error analysis for linear convection-diffusion problems under weak mesh regularity assumptions, Report 97/03, Oxford University Computing Laboratory, Oxford, UK, 1997. G. Knowles, Finite element approximation of parabolic time optimal control problems, SIAM J. Control Optim., 20 (1982), 414-427. J. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971. J. Lions and E. Magenes, Non homogeneous boundary value problems and applications, Grandlehre B. 181, Springer-Verlag, 1972. R. Li and W. Liu, http://circus.math.pku.edu.cn/AFEPack. W. Liu, H. Ma, T. Tang, and N. Yan, A posteriori error estimates for discontinuous Galerkin time-stepping method for optimal control problems governed by parabolic equations, SIAM J. Numer. Anal., 42 (2004), 1032-1061. W. Liu and N. Yan, A posteriori error estimates for convex boundary control problems, SIAMJ. Numer. Anal., 39 (2001), 73-99. W. Liu and N. Yan, A posteriori error estimates for optimal control problems governed by Stokes equations, SIAM J. Numer. Anal., 40 (2003), 1850-1869. R. Li, W. Liu, H. Ma, and T. Tang, Adaptive finite element approximation of elliptic control problems, SIAM J. Control Optim., 41 (2002), 1321-1349. C. Meyer and A. Rösch, Superconvergence properties of optimal control problems. SIAM J. Control Optim., 43:3 (2004), 970-985. D. Meidner and B. Vexler, 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 (2008), 1150-1177. D. Meidner and B. Vexler, A priori error estimates for space-time finite element discretization of parabolic optimal control problems Part I: Problems with control constraints, SIAM J. Control Optim., 47 (2008), 1301-1329. R. Mcknight andW. Bosarge, The Ritz-Galerkin procedure for parabolic control problems, SIAM J. Control Optim., 11 (1973), 510-524. P. Neittaanmäki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications, M. Dekker, New York, 1994. P.A. Raviart and J.M. Thomas, A mixed finite element method for 2nd order elliptic problems, Aspecs of the Finite Element Method, Lecture Notes in Math, Springer, Berlin., 606 (1977), 292-315. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, 2nd ed, Springer Ser. Comput. Math. 25, Springer-Verlag, Berlin, 1972. X. Xing and Y. Chen, Error estimates of mixed methods for optimal control problems governed by parabolic equations, Internat. J. Numer. Methods in Engineering., 75 (2008), 735-754.
  Junming Duan, Huazhong Tang. AN EFFICIENT ADER DISCONTINUOUS GALERKIN SCHEME FOR DIRECTLY SOLVING HAMILTON-JACOBI EQUATION [J]. Journal of Computational Mathematics, 2020, 38(1): 58-83.  Weijie Huang, Zhiping Li. A MIXED FINITE ELEMENT METHOD FOR MULTI-CAVITY COMPUTATION IN INCOMPRESSIBLE NONLINEAR ELASTICITY [J]. Journal of Computational Mathematics, 2019, 37(5): 609-628.  Ruo Li, Pingbing Ming, Zhiyuan Sun, Fanyi Yang, Zhijian Yang. A DISCONTINUOUS GALERKIN METHOD BY PATCH RECONSTRUCTION FOR BIHARMONIC PROBLEM [J]. Journal of Computational Mathematics, 2019, 37(4): 524-540.  Xinjie Dai, Aiguo Xiao. NUMERICAL SOLUTIONS OF NONAUTONOMOUS STOCHASTIC DELAY DIFFERENTIAL EQUATIONS BY DISCONTINUOUS GALERKIN METHODS [J]. Journal of Computational Mathematics, 2019, 37(3): 419-436.  Fei Wang, Tianyi Zhang, Weimin Han. C0 DISCONTINUOUS GALERKIN METHODS FOR A PLATE FRICTIONAL CONTACT PROBLEM [J]. Journal of Computational Mathematics, 2019, 37(2): 184-200.  Lina Dong, Shaochun Chen. UNIFORMLY CONVERGENT NONCONFORMING TETRAHEDRAL ELEMENT FOR DARCY-STOKES PROBLEM [J]. Journal of Computational Mathematics, 2019, 37(1): 130-150.  Fan Zhang, Tiegang Liu, Jian Cheng. HIGH ORDER STABLE MULTI-DOMAIN HYBRID RKDG AND WENO-FD METHODS [J]. Journal of Computational Mathematics, 2018, 36(4): 517-541.  Liang Ge, Tongjun Sun. A SPARSE GRID STOCHASTIC COLLOCATION AND FINITE VOLUME ELEMENT METHOD FOR CONSTRAINED OPTIMAL CONTROL PROBLEM GOVERNED BY RANDOM ELLIPTIC EQUATIONS [J]. Journal of Computational Mathematics, 2018, 36(2): 310-330.  Yao Cheng, Qiang Zhang. LOCAL ANALYSIS OF THE FULLY DISCRETE LOCAL DISCONTINUOUS GALERKIN METHOD FOR THE TIME-DEPENDENT SINGULARLY PERTURBED PROBLEM [J]. Journal of Computational Mathematics, 2017, 35(3): 265-288.  Mahboub Baccouch. OPTIMAL A POSTERIORI ERROR ESTIMATES OF THE LOCAL DISCONTINUOUS GALERKIN METHOD FOR CONVECTIONDIFFUSION PROBLEMS IN ONE SPACE DIMENSION [J]. Journal of Computational Mathematics, 2016, 34(5): 511-531.  Ruihan Guo, Liangyue Ji, Yan Xu. HIGH ORDER LOCAL DISCONTINUOUS GALERKIN METHODS FOR THE ALLEN-CAHN EQUATION: ANALYSIS AND SIMULATION [J]. Journal of Computational Mathematics, 2016, 34(2): 135-158.  Jian Cheng, Kun Wang, Tiegang Liu. A GENERAL HIGH-ORDER MULTI-DOMAIN HYBRID DG/WENO-FD METHOD FOR HYPERBOLIC CONSERVATION LAWS [J]. Journal of Computational Mathematics, 2016, 34(1): 30-48.  Carolina Domínguez, Gabriel N. Gatica, Salim Meddahi. A POSTERIORI ERROR ANALYSIS OF A FULLY-MIXED FINITE ELEMENT METHOD FOR A TWO-DIMENSIONAL FLUID-SOLID INTERACTION PROBLEM [J]. Journal of Computational Mathematics, 2015, 33(6): 606-641.  Yang Yang, Chi-Wang Shu. ANALYSIS OF SHARP SUPERCONVERGENCE OF LOCAL DISCONTINUOUS GALERKIN METHOD FOR ONE-DIMENSIONAL LINEAR PARABOLIC EQUATIONS [J]. Journal of Computational Mathematics, 2015, 33(3): 323-340.  Jun Hu. FINITE ELEMENT APPROXIMATIONS OF SYMMETRIC TENSORS ON SIMPLICIAL GRIDS IN Rn: THE HIGHER ORDER CASE [J]. Journal of Computational Mathematics, 2015, 33(3): 283-296.
Viewed
Full text

Abstract