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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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
Keywords
- Virtual element methods
- PDE-constrained optimization
- Boundary control
- Discretize-then-optimize
- Optimal control
- Error estimates
- Primal-dual algorithm
- Numerical experiments