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2023, Volume 31, Issue 8: 4818-4842. doi: 10.3934/era.2023247
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Two-grid methods of finite element approximation for parabolic integro-differential optimal control problems

  • 1.
    School of Mathematics, Jilin University, Jilin 130012, China
  • 2.
    School of Mathematics and Statistics, Beihua University, Jilin 132013, China
  • Received: 27 April 2023 Revised: 16 June 2023 Accepted: 25 June 2023 Published: 07 July 2023

  • In this paper, we present a two-grid scheme of fully discrete finite element approximation for optimal control problems governed by parabolic integro-differential equations. The state and co-state variables are approximated by a piecewise linear function and the control variable is discretized by a piecewise constant function. First, we derive the optimal a priori error estimates for all variables. Second, we prove the global superconvergence by using the recovery techniques. Third, we construct a two-grid algorithm and discuss its convergence. In the proposed two-grid scheme, the solution of the parabolic optimal control problem on a fine grid is reduced to the solution of the parabolic optimal control problem on a much coarser grid; additionally, the solution of a linear algebraic system on the fine grid and the resulting solution maintain an asymptotically optimal accuracy. Finally, we present a numerical example to verify the theoretical results.

    Citation: Changling Xu, Huilai Li. Two-grid methods of finite element approximation for parabolic integro-differential optimal control problems[J]. Electronic Research Archive, 2023, 31(8): 4818-4842. doi: 10.3934/era.2023247

    Related Papers:

  • In this paper, we present a two-grid scheme of fully discrete finite element approximation for optimal control problems governed by parabolic integro-differential equations. The state and co-state variables are approximated by a piecewise linear function and the control variable is discretized by a piecewise constant function. First, we derive the optimal a priori error estimates for all variables. Second, we prove the global superconvergence by using the recovery techniques. Third, we construct a two-grid algorithm and discuss its convergence. In the proposed two-grid scheme, the solution of the parabolic optimal control problem on a fine grid is reduced to the solution of the parabolic optimal control problem on a much coarser grid; additionally, the solution of a linear algebraic system on the fine grid and the resulting solution maintain an asymptotically optimal accuracy. Finally, we present a numerical example to verify the theoretical results.


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