Superconvergence for Optimal Control Problems Governed by Semi-linear Elliptic Equations

“…Convergence and super convergence for finite element approximation of linear or semi-linear elliptic optimization problems are investigated in [1,2] and [3,4], respectively.…”

Section: Introductionmentioning

confidence: 99%

Superconvergence of Variational Discretization for Bilinear Elliptic Optimization Problems

Tang

Hua

2017

Proceedings of the 2017 International Conference on Applied Mathematics, Modelling and Statistics Application (AMMSA 2017)

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Abstract-In this paper, variational discretization approximation of bilinear elliptic optimal control problems is considered. Firstly, we construct a variational discretization method for the bilinear elliptic optimal control problem with control constraints. Secondly, we derive the convergence of the approximation scheme. Thirdly, we analyze the superconvergence. Finally, we present a numerical example to conform our theoretical results.

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“…Among these numerical methods, finite element discretization of the state equation is widely applied, although other methods are also used. For the studies about convergence and superconvergence of finite element approximations for optimal control problems, see, for example, [2][3][4][5][6][7][8][9][10][11][12][13][14]. A systematic introduction of finite element methods for PDEs and optimal control problems can be found in, for example, [15,1].…”

Section: Introductionmentioning

confidence: 99%

A priori and a posteriori error estimates ofH1-Galerkin mixed finite element methods for elliptic optimal control problems

Hou

2015

Computers & Mathematics with Applications

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A priori error estimates A posteriori error estimates H 1 -Galerkin mixed finite element methods a b s t r a c tIn this paper, we investigate numerical approximations of H 1 -Galerkin mixed finite element methods for elliptic optimal control problems. The presented scheme is independent symmetric and positive definite for the state variables and the adjoint state variables. Moreover, the matching relation (i.e., LBB-condition) between the mixed element spaces V h and W h is not necessary, thus, we can choose the approximation spaces more flexibly. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the standard finite element spaces, the control variable is approximated by piecewise constant functions. We derive a priori and a posteriori error estimates for the control variable, the state variables and the adjoint state variables. Finally, some numerical examples are given to demonstrate the theoretical results about a priori error estimates.

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Superconvergence for Optimal Control Problems Governed by Semi-linear Elliptic Equations

Cited by 57 publications

References 19 publications

A posteriori error estimates of mixed finite element solutions for fourth order parabolic control problems

A posteriori error estimates of mixed finite element solutions for fourth order parabolic control problems

Superconvergence of Variational Discretization for Bilinear Elliptic Optimization Problems

A priori and a posteriori error estimates ofH1-Galerkin mixed finite element methods for elliptic optimal control problems

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