WEAK-DUALITY BASED ADAPTIVE FINITE ELEMENT METHODS FOR PDE-CONSTRAINED OPTIMIZATION WITH POINTWISE GRADIENT STATE-CONSTRAINTS

Adaptive finite element methods for optimization problems for second order linear el- liptic partial differential equations subject to pointwise constraints on the l2-norm of the gradient of the state are considered. In a weak duality setting, i.e. without assuming a constraint qualification such as...

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Bibliographic Details
Published inJournal of computational mathematics Vol. 30; no. 2; pp. 101 - 123
Main Authors Hintermüller, M., Hinze, Michael, Hoppe, Ronald H. W.
Format Journal Article
LanguageEnglish
Published Chinese Academy of Mathematices and System Sciences (AMSS) Chinese Academy of Sciences 01.03.2012
Subjects
A posteriori knowledge
Adjoints
Approximation
College mathematics
Error rates
Estimators
Finite element method
L2范数
Lagrange multipliers
Partial differential equations
偏微分方程
后验误差估计
弱对偶性
梯度
状态约束
约束优化问题
自适应有限元方法
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ISSN0254-9409
1991-7139
1991-7139
DOI10.4208/jcm.1109-m3522

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Summary:Adaptive finite element methods for optimization problems for second order linear el- liptic partial differential equations subject to pointwise constraints on the l2-norm of the gradient of the state are considered. In a weak duality setting, i.e. without assuming a constraint qualification such as the existence of a Slater point, residual based a posteriori error estimators are derived. To overcome the lack in constraint qualification on the continuous level, the weak Fenchel dual is utilized. Several numerical tests illustrate the performance of the proposed error estimators.Mathematics subject classification： 65N30, 90C46, 65N50, 49K20, 49N15, 65K10.
Bibliography:Adaptive finite element method, A posteriori errors, Dualization, Low regu-larity, Pointwise gradient constraints, State constraints, Weak solutions.
Adaptive finite element methods for optimization problems for second order linear el- liptic partial differential equations subject to pointwise constraints on the l2-norm of the gradient of the state are considered. In a weak duality setting, i.e. without assuming a constraint qualification such as the existence of a Slater point, residual based a posteriori error estimators are derived. To overcome the lack in constraint qualification on the continuous level, the weak Fenchel dual is utilized. Several numerical tests illustrate the performance of the proposed error estimators.Mathematics subject classification： 65N30, 90C46, 65N50, 49K20, 49N15, 65K10.
11-2126/O1
ISSN:0254-9409
1991-7139
1991-7139
DOI:10.4208/jcm.1109-m3522