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Two-grid Method of Expanded Mixed Finite Element Approximations for Parabolic Integro-differential Optimal Control Problems

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Abstract

This paper aims to construct a two-grid scheme of fully discretized expanded mixed finite element methods for optimal control problems governed by parabolic integro-differential equations and discuss a priori error estimates. The state variables and co-state variables are discretized by the lowest order Raviart-Thomas mixed finite element, and the control variable is approximated by piecewise constant functions. The time derivative is discretized by the backward Euler method. Firstly, we define some new mixed elliptic projections and prove the corresponding error estimates which play an important role in subsequent convergence analysis. Secondly, we derive a priori error estimates for all variables. Thirdly, we present a two-grid scheme and analyze its convergence. In the 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 and the solution of a decoupled linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy. At last, a numerical example is presented to verify the theoretical results.

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Authors and Affiliations

  1. School of Science, Nan**g University of Posts and Telecommunications, Nan**g, 210023, China

    Yan-** Chen

  2. School of Mathematics and Statistics, Linyi University, Linyi, 276005, China

    Jian-wei Zhou

  3. School of Mathematics and Statistics, Beihua University, Jilin, 132013, China

    Tian-liang Hou

Authors
  1. Yan-** Chen
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  2. Jian-wei Zhou
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  3. Tian-liang Hou
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Correspondence to Yan-** Chen.

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The authors declare no conflict of interest.

Additional information

The project is supported by the State Key Program of National Natural Science Foundation of China (No. 11931003) and Natural Science Research Start-up Foundation of Recruiting Talents of Nan**g University of Posts and Telecommunications (No. NY223127).

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Chen, Yp., Zhou, Jw. & Hou, Tl. Two-grid Method of Expanded Mixed Finite Element Approximations for Parabolic Integro-differential Optimal Control Problems. Acta Math. Appl. Sin. Engl. Ser. (2024). https://doi.org/10.1007/s10255-024-1099-2

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  • DOI: https://doi.org/10.1007/s10255-024-1099-2

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