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AIMS Mathematics

2022, Volume 7, Issue 4: 6153-6172. doi: 10.3934/math.2022342
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Research article

A two-grid $ P_0^2 $-$ P_1 $ mixed finite element scheme for semilinear elliptic optimal control problems

  • 1.
    School of Mathematics, Jilin University, 2699 Qianjin street, Changchun, 130012, Jilin, China
  • 2.
    School of Mathematics and Statistics, Beihua University, Jilin, 132013, Jilin, China
  • Received: 22 August 2021 Revised: 09 December 2021 Accepted: 23 December 2021 Published: 17 January 2022

  • MSC : 49J20, 65N30

  • This paper aims to construct a two-grid mixed finite element scheme for distributed optimal control governed by semilinear elliptic equations. The state and co-state are approximated by the $ P_0^2 $-$ P_1 $ pair and the control variable is approximated by the piecewise constant functions. First, a superclose result for the control variable and a priori error estimates for all variables are obtained. Second, a two-grid $ P_0^2 $-$ P_1 $ mixed finite element algorithm is presented and the corresponding error is analyzed. In the two-grid scheme, the solution of the semilinear elliptic optimal control problem on a fine grid is reduced to the solution of the semilinear elliptic optimal control problem on a much coarser grid and the solution of a linear decoupled algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy. We find that the two-grid method achieves the same convergence property as the $ P_0^2 $-$ P_1 $ mixed finite element method if the two mesh sizes satisfy $ h = H^2 $. Finally, a numerical example demonstrating our theoretical results is presented.

    Citation: Changling Xu, Hongbo Chen. A two-grid $ P_0^2 $-$ P_1 $ mixed finite element scheme for semilinear elliptic optimal control problems[J]. AIMS Mathematics, 2022, 7(4): 6153-6172. doi: 10.3934/math.2022342

    Related Papers:

  • This paper aims to construct a two-grid mixed finite element scheme for distributed optimal control governed by semilinear elliptic equations. The state and co-state are approximated by the $ P_0^2 $-$ P_1 $ pair and the control variable is approximated by the piecewise constant functions. First, a superclose result for the control variable and a priori error estimates for all variables are obtained. Second, a two-grid $ P_0^2 $-$ P_1 $ mixed finite element algorithm is presented and the corresponding error is analyzed. In the two-grid scheme, the solution of the semilinear elliptic optimal control problem on a fine grid is reduced to the solution of the semilinear elliptic optimal control problem on a much coarser grid and the solution of a linear decoupled algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy. We find that the two-grid method achieves the same convergence property as the $ P_0^2 $-$ P_1 $ mixed finite element method if the two mesh sizes satisfy $ h = H^2 $. Finally, a numerical example demonstrating our theoretical results is presented.


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