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Nitsche’s method for elliptic Dirichlet boundary control problems on curved domains

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Abstract

We consider Nitsche’s method for solving elliptic Dirichlet boundary control problems on curved domains with control constraints. By using Nitsche’s method for the treatment of inhomogeneous Dirichlet boundary conditions, the L2 boundary control enters in the variational formulation in a natural sense. The idea was first used in Chang, et al. (Math. Anal. Appl. 453, 529–557 2017) where the curved boundary was approximated by a broken line and a locally defined mapping was needed to obtain the numerical control on the curved boundary. In this paper, we develop a method defined on curved domains directly. We derive a priori estimates of quasi-optimal order for the control in the L2 norm, and quasi-optimal order for the state and adjoint state in energy norms. Numerical examples are provided to show the performance of the proposed method.

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Acknowledgements

The authors would like to thank the anonymous referees for beneficial comments and suggestions.

Funding

This work was partially supported by the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20200848), the National Natural Science Foundation of China (Grant Nos.12101327 and 11971241), and the Supporting Project of National Natural Science Youth Fund of Nanjing University of Chinese Medicine (Grant No. XPT12101327).

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Authors and Affiliations

  1. School of Artificial Intelligence and Information Technology, Nanjing University of Chinese Medicine, Nanjing, 210023, China

    Qian Zhang

  2. Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

    Zhiyue Zhang

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  1. Qian Zhang
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Correspondence to Qian Zhang.

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Zhang, Q., Zhang, Z. Nitsche’s method for elliptic Dirichlet boundary control problems on curved domains. Numer Algor 94, 511–545 (2023). https://doi.org/10.1007/s11075-023-01510-3

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