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

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 Ravia...

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Vydané v:	Numerical algorithms Ročník 96; číslo 2; s. 879 - 924
Hlavní autori:	Shakya, Pratibha, Kumar Sinha, Rajen
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.06.2024 Springer Nature B.V
Predmet:	Algebra Algorithms Approximation Computer Science Discretization Error analysis Estimates Estimators Finite element method Implicit methods Numeric Computing Numerical Analysis Optimal control Original Paper Spacetime Theory of Computation Variables The backward-Euler method 65N15 Mixed finite element method Variational discretization 49J20 A posteriori error estimates Parabolic optimal control problems 65N30
ISSN:	1017-1398, 1572-9265
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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 ∞ ( 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.
AbstractList	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∞(L2)-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. 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 ∞ ( 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.
Author	Kumar Sinha, Rajen Shakya, Pratibha
Author_xml	– sequence: 1 givenname: Pratibha surname: Shakya fullname: Shakya, Pratibha email: shakya.pratibha10@gmail.com organization: Department of Mathematics, Indian Institute of Science – sequence: 2 givenname: Rajen surname: Kumar Sinha fullname: Kumar Sinha, Rajen organization: Department of Mathematics, Indian Institute of Technology Guwahati
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References_xml	– reference: ClémentPApproximation by finite element function using local regularizationRAIRO Sér. Rouge Anal. Numér.197527784400739 – reference: TröltzschFOptimal control of partial differential equations, Theory2010Providence, RIMethods and Applications. AMS – reference: VerfürthRA posteriori error estimates for nonlinear problems: Lρ(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\rho }({\Omega })$$\end{document}-error estimates for finite element discretization of parabolic equationsMath. Comp.199867133513601604371 – reference: MenonSNatarajNPaniAKAn a posteriori error analysis of a mixed finite element Galerkin approximation to second order linear parabolic problemsSIAM J. Numer. 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Math.198240373406695603 – reference: JohnsonCThoméeVError estimates for some mixed finite elements methods for parabolic type problemsRAIRO Analyse Numerique.1981154178610597 – reference: LuZNew a posteriori L∞(L2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty (L^2)$$\end{document} and L2(L2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2(L^2)$$\end{document}-error estimates of mixed finite element methods for general nonlinear parabolic optimal control problemsAppl. Math.2016611351633470771 – reference: Neittaanmäki, P.: Tiba, D.: Optimal control of nonlinear parabolic systems: theory, algorithms and applications, Dekker: New York (1994) – reference: ManoharRSinhaRKA posteriori L∞(L∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\infty }(L^{\infty })$$\end{document} error estimates for finite element approximations to parabolic optimal control problemsComput. Appl. Math.2021402984338780 – reference: LasieckaIRitz-Galerkin approximation of the time optimal boundary control problem for parabolic systems with Dirichlet boundary conditionsSIAM J. Control Optim.198422477500739837 – reference: HinzeMA variational discretization concept in control constrained optimization: the linear quadratic caseComput. Optim. 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Snippet	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...
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SubjectTerms	Algebra Algorithms Approximation Computer Science Discretization Error analysis Estimates Estimators Finite element method Implicit methods Numeric Computing Numerical Analysis Optimal control Original Paper Spacetime Theory of Computation Variables
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Title	Analysis for the space-time a posteriori error estimates for mixed finite element solutions of parabolic optimal control problems
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