Approximation of an elliptic control problem by the finite element method

French, Donald A.; King, J. Thomas

doi:10.1080/01630569108816430

Cited by 73 publications

(35 citation statements)

References 5 publications

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“…The method of transposition goes back at least to Lions and Magenes, [16], and is used by several other authors including French and King, [12], Casas and Raymond, [6], Deckelnick, Günther, and Hinze, [11], and May, Rannacher, and Vexler, [17]. Since by partial integration the derivation…”

Section: Methods Of Transposition In Convex Polygonal Domainsmentioning

confidence: 99%

“…We assume Ω ⊂ R 2 to be a bounded polygonal domain with boundary Γ. Such problems arise in optimal control when the Dirichlet boundary control is considered in L 2 (Γ) only, see for example the papers by Deckelnick, Günther, and Hinze, [11], French and King, [12], May, Rannacher, and Vexler, [17], and Apel, Mateos, Pfefferer, and Rösch, [1]. On the continuous level we even admit more irregular data.…”

Section: Introductionmentioning

confidence: 99%

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Discretization of the Poisson equation with non-smooth data and emphasis on non-convex domains

Apel

Nicaise

Pfefferer

2016

Numer. Methods Partial Differential Eq.

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Several approaches are discussed how to understand the solution of the Dirichlet problem for the Poisson equation when the Dirichlet data are non-smooth such as if they are in L 2 only. For the method of transposition (sometimes called very weak formulation) three spaces for the test functions are considered, and a regularity result is proved. An approach of Berggren is recovered as the method of transposition with the second variant of test functions. A further concept is the regularization of the boundary data combined with the weak solution of the regularized problem. The effect of the regularization error is studied.The regularization approach is the simplest to discretize. The discretization error is estimated for a sequence of quasi-uniform meshes. Since this approach turns out to be equivalent to Berggren's discretization his error estimates are rendered more precisely. Numerical tests show that the error estimates are sharp, in particular that the order becomes arbitrarily small when the maximal interior angle of the domain tends to 2π.Key Words Elliptic boundary value problem, method of transposition, very weak formulation, finite element method, discretization error estimate

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Section: Methods Of Transposition In Convex Polygonal Domainsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Discretization of the Poisson equation with non-smooth data and emphasis on non-convex domains

Apel

Nicaise

Pfefferer

2016

Numer. Methods Partial Differential Eq.

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“…Finite element approximation of optimal control problems has long been an important topic in engineering design work and has been extensively studied in the literature. There have been extensive theoretical and numerical studies for finite element approximation of various optimal control problems; see [2,12,13,15,20,23,37,44]. For instance, for the optimal control problems governed by some linear elliptic or parabolic state equations, a priori error estimates of the finite element approximation were established long ago; see, for example, [12,13,15,20,23,37].…”

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confidence: 99%

Adaptive Finite Element Approximation for Distributed Elliptic Optimal Control Problems

Liu

et al. 2002

SIAM J. Control Optim.

246

126

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Abstract. In this paper, sharp a posteriori error estimators are derived for a class of distributed elliptic optimal control problems. These error estimators are shown to be useful in adaptive finite element approximation for the optimal control problems and are implemented in the adaptive approach. Our numerical results indicate that the sharp error estimators work satisfactorily in guiding the mesh adjustments and can save substantial computational work.Key words. mesh adaptivity, optimal control, a posteriori error estimate, finite element method AMS subject classifications. 49J20, 65N30PII. S0363012901389342 1. Introduction. Finite element approximation of optimal control problems has long been an important topic in engineering design work and has been extensively studied in the literature. There have been extensive theoretical and numerical studies for finite element approximation of various optimal control problems; see [2,12,13,15,20,23,37,44]. For instance, for the optimal control problems governed by some linear elliptic or parabolic state equations, a priori error estimates of the finite element approximation were established long ago; see, for example, [12,13,15,20,23,37]. Furthermore, a priori error estimates were established for the finite element approximation of some important flow control problems in [17] and [11]. A priori error estimates have also been obtained for a class of state constrained control problems in [43], though the state equation is assumed to be linear. In [29], this assumption has been removed by reformulating the control problem as an abstract optimization problem in some Banach spaces and then applying nonsmooth analysis. In fact, the state equation there can be a variational inequality.In recent years, the adaptive finite element method has been extensively investigated. Adaptive finite element approximation is among the most important means to boost the accuracy and efficiency of finite element discretizations. It ensures a higher density of nodes in a certain area of the given domain, where the solution is more difficult to approximate. At the heart of any adaptive finite element method is an a posteriori error estimator or indicator. The literature in this area is extensive. Some of the techniques directly relevant to our work can be found in [1,5,6,7,28,32,34,42,47]. It is our belief that adaptive finite element enhancement is one of the future directions to pursue in developing sophisticated numerical methods for optimal design problems.

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“…G Remark. The function û and the argument used in the proof of Lemma 7 were introduced in French and King [11]. Also, see the related work in Fix, Gunzburger, and Peterson [10].…”

Section: Smooth Solution Estimatesmentioning

confidence: 99%

Analysis of a Robust Finite Element Approximation for a Parabolic Equation with Rough Boundary Data

French¹,

King²

1993

Mathematics of Computation

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Abstract.The approximation of parabolic equations with nonhomogeneous Dirichlet boundary data by a numerical method that consists of finite elements for the space discretization and the backward Euler time discretization is studied. The boundary values are assumed in a least squares sense. It is shown that this method achieves an optimal rate of convergence for rough (only L1) boundary data and for smooth data as well. The results of numerical computations which confirm the robust theoretical error estimates are also presented.

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Approximation of an elliptic control problem by the finite element method

Cited by 73 publications

References 5 publications

Discretization of the Poisson equation with non-smooth data and emphasis on non-convex domains

Discretization of the Poisson equation with non-smooth data and emphasis on non-convex domains

Adaptive Finite Element Approximation for Distributed Elliptic Optimal Control Problems

Analysis of a Robust Finite Element Approximation for a Parabolic Equation with Rough Boundary Data

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