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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
  • Received:2012-09-23 Revised:2014-11-17 Online:2015-03-15 Published:2015-03-13
  • 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.

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