Superconvergence property of finite element methods for parabolic optimal control problems

Hou, Chunjuan; Chen, Yanping; Lu, Zuliang

doi:10.3934/jimo.2011.7.927

Cited by 9 publications

(6 citation statements)

References 33 publications

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“…The convergent order will be improved to O(h 3 2 ) or O(h 3 2 + k) by superconvergence analysis. Some superconvergence results of FEMs for linear and semilinear elliptic or parabolic OCPs can be found in [4,6,15,[27][28][29]. Adaptive FEMs that approximate elliptic and parabolic OCPs have been investigated in [1,11,19,32] and [3], respectively.…”

Section: Introductionmentioning

confidence: 99%

Convergence and superconvergence of variational discretization for parabolic bilinear optimization problems

Tang¹,

Hua²

2019

J Inequal Appl

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In this paper, we investigate a variational discretization approximation of parabolic bilinear optimal control problems with control constraints. For the state and co-state variables, triangular linear finite element and difference methods are used for space and time discretization, respectively, superconvergence in H 1 -norm between the numerical solutions and elliptic projections are derived. Although the control variable is not discrete directly, convergence of second order in L 2 -norm is obtained. These theoretical results are confirmed by two numerical examples.

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Section: Introductionmentioning

confidence: 99%

Convergence and superconvergence of variational discretization for parabolic bilinear optimization problems

Tang¹,

Hua²

2019

J Inequal Appl

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“…The superconvergence of nonlinear parabolic problem was studied in [19]. In [20], superconvergence was obtained for parabolic optimal control problems with convex control constraints, where the state partial differential equations are linear.…”

Section: Introductionmentioning

confidence: 99%

Superconvergence for General Convex Optimal Control Problems Governed by Semilinear Parabolic Equations

Dai

Chen

2014

ISRN Applied Mathematics

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We will investigate the superconvergence for the semidiscrete finite element approximation of distributed convex optimal control problems governed by semilinear parabolic equations. The state and costate are approximated by the piecewise linear functions and the control is approximated by piecewise constant functions. We present the superconvergence analysis for both the control variable and the state variables.

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“…For semilinear elliptic optimal control problems, a priori error estimates were investigated in [1], a posteriori error estimates have been obtained in [11], a superconvergence can be seen in [3], Borzì considered a second-order discretization and multigrid solution in [2]. Recently, some adaptive algorithm and superconvergence results can be found in [6,7,15]. Notice that all the control variables are first-order convergence and the superconvergence of the state and the adjoint state variables are 1.5 order in the above works.…”

Section: Introductionmentioning

confidence: 99%

An Improved Finite Element Approximation and Superconvergence for Temperature Control Problems

Tang

2013

Mathematical Modelling and Analysis

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In this paper, we consider an improved finite element approximation for temperature control problems, where the state and the adjoint state are discretized by piecewise linear functions while the control is not discretized directly. The numerical solution of the control is obtained by a projection of the adjoint state to the set of admissible controls. We derive a priori error estimates and superconvergence of second-order. Moreover, we present some numerical examples to illustrate our theoretical results.

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Superconvergence property of finite element methods for parabolic optimal control problems

Cited by 9 publications

References 33 publications

Convergence and superconvergence of variational discretization for parabolic bilinear optimization problems

Convergence and superconvergence of variational discretization for parabolic bilinear optimization problems

Superconvergence for General Convex Optimal Control Problems Governed by Semilinear Parabolic Equations

An Improved Finite Element Approximation and Superconvergence for Temperature Control Problems

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