Error estimates and superconvergence of mixed covolume approximations for elliptic optimal control problems

Chunjuan Hou; Yanping Chen; Jian Huang; Jiawang Liu; Fangfang Qin; Chunjuan Hou; Yanping Chen; Jian Huang; Jiawang Liu; Fangfang Qin

doi:10.3934/jimo.2026030

Journal of Industrial and Management Optimization

2026, Volume 22, Issue 2: 832-859. doi: 10.3934/jimo.2026030

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Research article

Error estimates and superconvergence of mixed covolume approximations for elliptic optimal control problems

1.
National Center for Applied Mathematics in Hunan, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China
2.
School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China

Received: 25 September 2025 Revised: 10 December 2025 Accepted: 18 December 2025 Published: 05 January 2026
49J20, 65N30

In this scholarly article, we analyze the results of error estimates and phenomena of superconvergence associated with the mixed covolume approximation method, which is applied to a particular class of linear elliptic optimal control problems. The control variable is discretized using piecewise constant functions. Additionally, the state and the costate variables are both approximated using the lowest-order Raviart–Thomas ($ RT_0 $) mixed finite element method. First, mixed covolume approximation of optimal control problems is constructed. Second, "a priori error estimations" for each variable are computed. Third, a superconvergence result is established, and it is proved that there exists a second-order superconvergence relationship between the centroid interpolation of variable $ u $ and its numerical solution. Finally, two carefully designed numerical examples are presented to validate the reliability of the theoretical findings, providing concrete evidence to corroborate the above results and strengthen the coherence of the study's conclusions.
- mixed covolume method,
- superconvergence,
- elliptic equations,
- optimal control problems,
- error estimates
Citation: Chunjuan Hou, Yanping Chen, Jian Huang, Jiawang Liu, Fangfang Qin. Error estimates and superconvergence of mixed covolume approximations for elliptic optimal control problems[J]. Journal of Industrial and Management Optimization, 2026, 22(2): 832-859. doi: 10.3934/jimo.2026030

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Abstract

In this scholarly article, we analyze the results of error estimates and phenomena of superconvergence associated with the mixed covolume approximation method, which is applied to a particular class of linear elliptic optimal control problems. The control variable is discretized using piecewise constant functions. Additionally, the state and the costate variables are both approximated using the lowest-order Raviart–Thomas ($ RT_0 $) mixed finite element method. First, mixed covolume approximation of optimal control problems is constructed. Second, "a priori error estimations" for each variable are computed. Third, a superconvergence result is established, and it is proved that there exists a second-order superconvergence relationship between the centroid interpolation of variable $ u $ and its numerical solution. Finally, two carefully designed numerical examples are presented to validate the reliability of the theoretical findings, providing concrete evidence to corroborate the above results and strengthen the coherence of the study's conclusions.

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