Abstract
In this paper, we propose a new mixed finite element method, called stabilized mixed finite element method, for the approximation of optimal control problems constrained by a first-order elliptic system. This method is obtained by adding suitable elementwise least-squares residual terms for the primal state variable y and its flux \(\sigma \). We prove the coercive and continuous properties for the new mixed bilinear formulation at both continuous and discrete levels. Therefore, the finite element function spaces do not require to satisfy the Ladyzhenkaya–Babuska–Brezzi consistency condition. Furthermore, the state and flux state variables can be approximated by the standard Lagrange finite element. We derive optimality conditions for such optimal control problems under the concept of Discretization-then-Optimization, and then a priori error estimates in a weighted norm are discussed. Finally, numerical experiments are given to confirm the efficiency and reliability of the stabilized method.
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Acknowledgments
The authors would like to thank the editor and the anonymous referee for their valuable comments and suggestions on an earlier version of this paper. This work was supported by the National Natural Science Foundation of China (Nos. 11201485, 91330106), the Promotive Research Fund for Excellent Young and Middle-aged Scientists of Shandong Province (No. BS2013NJ001), the Fundamental Research Funds for the Central Universities (Nos. 14CX02217A, 15CX08004A).
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Fu, H., Rui, H., Hou, J. et al. A Stabilized Mixed Finite Element Method for Elliptic Optimal Control Problems. J Sci Comput 66, 968–986 (2016). https://doi.org/10.1007/s10915-015-0050-3
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DOI: https://doi.org/10.1007/s10915-015-0050-3
Keywords
- Optimal control
- Stabilized mixed finite element
- LBB condition
- A priori error estimates
- Numerical experiments