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Analysis for the space-time a posteriori error estimates for mixed finite element solutions of parabolic optimal control problems

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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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The authors do not analyze or generate any datasets, because our work proceeds within a theoretical and mathematical approach. One can obtain the relevant materials from the references below.

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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.

Funding

Under the scheme of National Post-Doctoral Fellowship (PDF/2021/000444) by the Department of Science and Technology (DST), India.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Science, Bangalore, 560012, India

    Pratibha Shakya

  2. Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, 781039, India

    Rajen Kumar Sinha

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  1. Pratibha Shakya
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Correspondence to Pratibha Shakya.

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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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