Superconvergence for optimal control problems governed by semilinear parabolic equations

Chunjuan Hou; Zuliang Lu; Xuejiao Chen; Xiankui Wu; Fei Cai; Chunjuan Hou; Zuliang Lu; Xuejiao Chen; Xiankui Wu; Fei Cai

doi:10.3934/math.2022522

AIMS Mathematics

2022, Volume 7, Issue 5: 9405-9423. doi: 10.3934/math.2022522

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Superconvergence for optimal control problems governed by semilinear parabolic equations

1.
Department of Data Science, Guangzhou Huashang College, Guangzhou 511300, China
2.
Key Laboratory for Nonlinear Science and System Structure, Chongqing Three Gorges University, Chongqing, 404000, China
3.
Research Center for Mathematics and Economics, Tianjin University of Finance and Economics, Tianjin, 300222, China

Received: 08 October 2021 Revised: 03 February 2022 Accepted: 28 February 2022 Published: 11 March 2022
MSC : 49J20, 65N30

In this paper, we first investigate optimal control problem for semilinear parabolic and introduce the standard $ L^2(\Omega) $-orthogonal projection and the elliptic projection. Then we present some necessary intermediate variables and their error estimates. At last, we derive the error estimates between the finite element solutions and $ L^2 $-orthogonal projection or the elliptic projection of the exact solutions.
- finite element approximation,
- semilinear parabolic equation,
- optimal control problem,
- the elliptic projection,
- $ L^2 $-orthogonal projection,
- superconvergence
Citation: Chunjuan Hou, Zuliang Lu, Xuejiao Chen, Xiankui Wu, Fei Cai. Superconvergence for optimal control problems governed by semilinear parabolic equations[J]. AIMS Mathematics, 2022, 7(5): 9405-9423. doi: 10.3934/math.2022522

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Abstract

In this paper, we first investigate optimal control problem for semilinear parabolic and introduce the standard $ L^2(\Omega) $-orthogonal projection and the elliptic projection. Then we present some necessary intermediate variables and their error estimates. At last, we derive the error estimates between the finite element solutions and $ L^2 $-orthogonal projection or the elliptic projection of the exact solutions.

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