Abstract
This paper investigates the space-time residual-based a posteriori error bounds of the mixed finite element method for the optimal control problem governed by the parabolic equation in a bounded convex domain. For the spatial discretization of the state and co-state variables, the lowest-order Raviart-Thomas spaces are utilized, although for the control variable, variational discretization technique is used. The backward-Euler implicit method is applied for temporal discretization. To provide a posteriori error estimates for the state and control variables in the \(L^{\infty }(L^2)\)-norm, an elliptic reconstruction approach paired with an energy strategy is utilized. The reliability and efficiency of the a posteriori error estimators are discussed. The effectiveness of the estimators is finally confirmed through the numerical tests.
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Acknowledgements
The first author would like to thank the Department of Science and Technology (DST) for providing financial assistance under the scheme National Post-Doctoral Fellowship (PDF/2021/000444), New Delhi, India. Finally, the authors acknowledge the valuable comments and constructive suggestions made by the referees that led to the improvement of the content of the manuscript.
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Under the scheme of National Post-Doctoral Fellowship (PDF/2021/000444) by the Department of Science and Technology (DST), India.
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Shakya, P., Kumar Sinha, R. Analysis for the space-time a posteriori error estimates for mixed finite element solutions of parabolic optimal control problems. Numer Algor 96, 879–924 (2024). https://doi.org/10.1007/s11075-023-01669-9
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DOI: https://doi.org/10.1007/s11075-023-01669-9
Keywords
- A posteriori error estimates
- Mixed finite element method
- Parabolic optimal control problems
- The backward-Euler method
- Variational discretization