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Virtual Element Method for Control Constrained Dirichlet Boundary Control Problem Governed by the Diffusion Problem

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Abstract

This article develops a conforming virtual element method for a control-constrained Dirichlet boundary optimal control problem governed by the diffusion problem. An energy-based cost functional is used to approximate the control problem which results in a smooth control in contrast to the \(L^2(\Gamma )\) approach which can lead to a control with discontinuities at the corners (Gong in SIAM J Numer Anal 60:450-474, 2022) . We use virtual element discretization of control, state, and adjoint variables along with a discretize-then-optimize approach to compute the optimal control is used to solve the problem. A new framework for the a priori error analysis is presented, which is optimal up to the regularity of the continuous solution. A primal-dual algorithm is used to solve the Dirichlet optimal control problem, and numerical experiments are conducted to illustrate the theoretical findings on general polygonal meshes.

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Acknowledgements

The third author is supported by the Department of Science and Technology (DST-SERB) India (grant number CRG/2019/003863). The first author would like to thank Dr. Rekha Khot for a helpful discussion and acknowledges BITS-Pilani, K K Birla Goa Campus where the majority of this work was carried out during his stay there.

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Authors and Affiliations

  1. School of Mathematics, Monash University, Melbourne, Australia

    Jai Tushar

  2. Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, China

    Ramesh Chandra Sau

  3. Department of Mathematics, BITS Pilani-KK Birla Goa Campus, Zuarinagar, Goa, India

    Anil Kumar

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  1. Jai Tushar
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  2. Ramesh Chandra Sau
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  3. Anil Kumar
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Correspondence to Jai Tushar.

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Tushar, J., Sau, R.C. & Kumar, A. Virtual Element Method for Control Constrained Dirichlet Boundary Control Problem Governed by the Diffusion Problem. J Sci Comput 98, 21 (2024). https://doi.org/10.1007/s10915-023-02410-3

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  • DOI: https://doi.org/10.1007/s10915-023-02410-3

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