Error estimates of mixed finite elements combined with Crank-Nicolson scheme for parabolic control problems

Yuelong Tang; Yuelong Tang

doi:10.3934/math.2023628

AIMS Mathematics

2023, Volume 8, Issue 5: 12506-12519. doi: 10.3934/math.2023628

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Research article

Error estimates of mixed finite elements combined with Crank-Nicolson scheme for parabolic control problems

Yuelong Tang ^,

Institute of Computational Mathematics, College of Science, Hunan University of Science and Engineering, Yongzhou 425100, Hunan, China

Received: 06 February 2023 Revised: 14 March 2023 Accepted: 20 March 2023 Published: 27 March 2023
MSC : 49J20, 65N22, 65N30

In this paper, a mixed finite element method combined with Crank-Nicolson scheme approximation of parabolic optimal control problems with control constraint is investigated. For the state and co-state, the order $ m = 1 $ Raviart-Thomas mixed finite element spaces and Crank-Nicolson scheme are used for space and time discretization, respectively. The variational discretization technique is used for the control variable. We derive optimal priori error estimates for the control, state and co-state. Some numerical examples are presented to demonstrate the theoretical results.
- parabolic optimal control problems,
- Crank-Nicolson scheme,
- mixed finite element method,
- variational discretization,
- a priori error estimates
Citation: Yuelong Tang. Error estimates of mixed finite elements combined with Crank-Nicolson scheme for parabolic control problems[J]. AIMS Mathematics, 2023, 8(5): 12506-12519. doi: 10.3934/math.2023628

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Abstract

In this paper, a mixed finite element method combined with Crank-Nicolson scheme approximation of parabolic optimal control problems with control constraint is investigated. For the state and co-state, the order $ m = 1 $ Raviart-Thomas mixed finite element spaces and Crank-Nicolson scheme are used for space and time discretization, respectively. The variational discretization technique is used for the control variable. We derive optimal priori error estimates for the control, state and co-state. Some numerical examples are presented to demonstrate the theoretical results.

References

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