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Mixed Virtual Element Approximation of a Fourth Order Optimal Control Problem

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Abstract

In this article, we study mixed virtual element methods for a distributed optimal control problem governed by a fourth order partial differential equation. By introducing an auxiliary variable, the fourth order equation can be transformed into systems of second order equations. A mixed virtual element discrete scheme for the optimal control problem is established. Moreover, a priori error estimates for auxiliary, state, adjoint state and control variable in \({{H}^{1}}\) and \({{L}^{2}}\) norms are derived. Finally, the theoretical finding is verified by numerical experiments.

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Funding

Zhaojie Zhou was supported by the National Natural Science Foundation of China under Grant nos. 11971276, 12171287. Yue Shen was supported by the Natural Science Foundation of Shanxi Province (Grant no. 2021JQ-492).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, 250014, Ji’nan, China

    Minghui Yang & Zhaojie Zhou

  2. School of Science, Xi’an University of Architecture and Technology, 710055, Xi’an, China

    Yue Shen

Authors
  1. Minghui Yang
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  2. Yue Shen
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  3. Zhaojie Zhou
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Correspondence to Zhaojie Zhou.

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Yang, M., Shen, Y. & Zhou, Z. Mixed Virtual Element Approximation of a Fourth Order Optimal Control Problem. Comput. Math. and Math. Phys. 63, 1001–1015 (2023). https://doi.org/10.1134/S0965542523060180

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  • DOI: https://doi.org/10.1134/S0965542523060180

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