CONVERGENCE AND QUASI-OPTIMALITY OF L2-NORMS BASED AN ADAPTIVE FINITE ELEMENT METHOD FOR NONLINEAR OPTIMAL CONTROL PROBLEMS

被引：1

作者：

Lu, Zuliang ^{[1
,2
]}

Huang, Fei ^{[1
]}

Wu, Xiankui ^{[1
]}

Li, Lin ^{[3
]}

Liu, Shang ^{[4
]}

机构：

[1] Chongqing Three Gorges Univ, Key Lab Nonlinear Sci & Syst Struct, Chongqing 404000, Peoples R China

[2] Tianjin Univ Finance & Econ, Res Ctr Math & Econ, Tianjin 300222, Peoples R China

[3] Chongqing Univ Posts & Telecommun, Coll Comp Sci & Technol, Chongqing 400065, Peoples R China

[4] Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410114, Hunan, Peoples R China

来源：

ELECTRONIC RESEARCH ARCHIVE | 2020年 / 28卷 / 04期

基金：

中国国家社会科学基金;

关键词：

Nonlinear optimal control problem; adaptive finite element method; convergence; quasi-optimality; APPROXIMATION; ALGORITHM;

D O I：

10.3934/era.2020077

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

This paper aims at investigating the convergence and quasi-optimality of an adaptive finite element method for control constrained nonlinear elliptic optimal control problems. We derive a posteriori error estimation for both the control, the state and adjoint state variables under controlling by L-2-norms where bubble function is a wonderful tool to deal with the global lower error bound. Then a contraction is proved before the convergence is proposed. Furthermore, we find that if keeping the grids sufficiently mildly graded, we can prove the optimal convergence and the quasi-optimality for the adaptive finite element method. In addition, some numerical results are presented to verify our theoretical analysis.

引用

页码：1459 / 1486

页数：28

共 32 条

[1] A posteriori error estimation in finite element analysis [J].

Ainsworth, M ;

Oden, JT .

COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING, 1997, 142 (1-2) :1-88

[2]

[Anonymous], TRENDS PDE CONSTRAIN

[3] ERROR ESTIMATES FOR ADAPTIVE FINITE-ELEMENT COMPUTATIONS [J].

BABUSKA, I ;

RHEINBOLDT, WC .

SIAM JOURNAL ON NUMERICAL ANALYSIS, 1978, 15 (04) :736-754

[4] A CONVERGENT NONCONFORMING ADAPTIVE FINITE ELEMENT METHOD WITH QUASI-OPTIMAL COMPLEXITY [J].

Becker, Roland ;

Mao, Shipeng ;

Shi, Zhongci .

SIAM JOURNAL ON NUMERICAL ANALYSIS, 2010, 47 (06) :4639-4659

[5] Adaptive finite element methods with convergence rates [J].

Binev, P ;

Dahmen, W ;

DeVore, R .

NUMERISCHE MATHEMATIK, 2004, 97 (02) :219-268

[6] Convergence analysis of a conforming adaptive finite element method for an obstacle problem [J].

Braess, Dietrich ;

Carstensen, Carsten ;

Hoppe, Ronald H. W. .

NUMERISCHE MATHEMATIK, 2007, 107 (03) :455-471

[7] Error reduction and convergence for an adaptive mixed finite element method [J].

Carstensen, Carsten ;

Hoppe, R. H. W. .

MATHEMATICS OF COMPUTATION, 2006, 75 (255) :1033-1042

[8] Error estimates for the numerical approximation of semilinear elliptic control problems with finitely many state constraints [J].

Casas, E .

ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS, 2002, 8 :345-374

[9] Quasi-optimal convergence rate for an adaptive finite element method [J].

Cascon, J. Manuel ;

Kreuzer, Christian ;

Nochetto, Ricardo H. ;

Siebert, Kunibert G. .

SIAM JOURNAL ON NUMERICAL ANALYSIS, 2008, 46 (05) :2524-2550

[10] Adaptive Finite Element Approximations for a Class of Nonlinear Eigenvalue Problems in Quantum Physics [J].

Chen, Huajie ;

Gong, Xingao ;

He, Lianhua ;

Zhou, Aihui .

ADVANCES IN APPLIED MATHEMATICS AND MECHANICS, 2011, 3 (04) :493-518

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