This paper presents a numerical method for the optimal control problem governed by the heat diffusion equation inside a composite medium. The contact resistance at the interface of constitute materials allows for jumps of the temperature field. The derivation process of the Karush-Kuhn-Tucher system is given by the formal Lagrange method. Due to the discontinuity of the temperature field, the standard linear finite element method cannot achieve optimal convergence when the uniform mesh is used. Therefore, the unfitted finite element method is applied to discrete the state equation required in the variational discretization approach. Optimal error estimates in the broken H1-norm and L2-norm for the control, state, and adjoint state are derived. Some numerical examples are provided to confirm the theoretical results.
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We are very grateful to anonymous referees for their valuable suggestions which have helped to improve the paper.
This work is partially supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 17KJB110014), the National Natural Science Foundation of China (Grant Nos. 11471166 and 11701291), and the Natural Science Foundation of Jiangsu Province (Grant No. BK20160880).
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Zhang, Q., Zhao, T. & Zhang, Z. Unfitted finite element for optimal control problem of the temperature in composite media with contact resistance. Numer Algor 84, 165–180 (2020). https://doi.org/10.1007/s11075-019-00750-6
- PDE-constrained optimization
- Interface problem
- Unfitted mesh