Finite element approximation of elliptic control problems with constraints on the gradient

Deckelnick, Klaus; Günther, Andreas; Hinze, Michael

doi:10.1007/s00211-008-0185-3

Cited by 66 publications

(78 citation statements)

References 12 publications

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“…on Γ where P [a,b] denotes the pointwise projection onto the interval [a, b]. In [3] we prove Lemma 1.1 Let u ∈ U ad be the solution of (2) with corresponding state y and adjoint state p. Then…”

Section: The Optimal Control Problemmentioning

confidence: 92%

“…To compare the discrete and continuous solutions we have to introduce a homeomorphism G h : Ω h → Ω with its restriction g h := G h|Γ h to the boundary Γ h := ∂Ω h , see [3]. Let us further define the space of piecewise linear finite elements X h := {v h ∈ C 0 (Ω) | v h|T ∈ P 1 (T ) ∀ T ∈ T h } and let γX h be the restriction to Γ h of functions in X h .…”

Section: A Priori Error Analysismentioning

confidence: 99%

“…The following theorem is proved in [3]: Theorem 2.1 Let (u, y) and (u h , y h ) be the solutions of (2) and (4).…”

Section: A Priori Error Analysismentioning

confidence: 99%

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A priori estimates for optimal Dirichlet boundary control problems

Deckelnick

Günther

Hinze

2009

Proc Appl Math and Mech

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We consider variational discretization of control constrained elliptic Dirichlet boundary control problems on smooth two-and three-dimensional domains, where we take into account the domain approximation. The state is discretized by linear finite elements, while the control variable is not discretized. We obtain optimal error bounds for the optimal control in two and three space dimensions. Furthermore we prove a superconvergence result in two space dimensions under the assumption that the underlying finite element meshes satisfy certain regularity requirements. We confirm our findings by a numerical experiment.Copyright line will be provided by the publisher The optimal control problemLet Ω ⊂ R d (d = 2, 3) be a bounded domain with a smooth boundary Γ := ∂Ω. Given f ∈ L 2 (Ω), u ∈ L 2 (Γ) we consider the boundary value problemwhere c ≥ 0 is a smooth function onΩ. It is well-known that (1) has a unique solution y ∈ H 1 2 (Ω) in a suitable weak sense which we denote by y = G(u).Next, let α > 0 and y 0 ∈ W 1,r (Ω),r > d be given. We then consider the Dirichlet boundary control problemwhere U ad = {u ∈ L 2 (Γ) | a ≤ u ≤ b a.e. on Γ} and a, b ∈ R, a < b. Existence of a unique solution u ∈ U ad of (2) follows by standard arguments. This solution is characterized by the variational inequalityLet us introduce the adjoint state p ∈ H 2 (Ω) ∩ H 1 0 (Ω) as the solution of the boundary value problemThe optimal control u is then given by u = P [a,b] 1 α ∂ ν p a.e. on Γ where P [a,b] denotes the pointwise projection onto the interval [a, b]. In [3] we prove Lemma 1

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Section: The Optimal Control Problemmentioning

confidence: 92%

Section: A Priori Error Analysismentioning

confidence: 99%

See 1 more Smart Citation

A priori estimates for optimal Dirichlet boundary control problems

Deckelnick

Günther

Hinze

2009

Proc Appl Math and Mech

Self Cite

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“…It is well known that (3.2)-(3.5) can be rewritten equivalently as a system of semi-smooth equations and thus can be solved by a semi-smooth Newton method, see for instance [4], [6], [12]. In particular, we can avoid the use of relaxation methods such as Moreau-Yosida relaxation, interior point methods, or Lavrentiev-type regularization.…”

Section: Remarkmentioning

confidence: 99%

“…The numerical solution of the corresponding systems (3.2)-(3.5) is performed with the semismooth Newton method proposed in [4], whose extension to the treatment of finite element approximations of semilinear PDEs ist straightforward. All the computations are performed on a uniform triangulation ofΩ with mesh size h = 2 −5 √ 2.…”

Section: Numerical Examplesmentioning

confidence: 99%

Global minima for semilinear optimal control problems

2016

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Abstract:We consider an optimal control problem subject to a semilinear elliptic PDE together with its variational discretization. We provide a condition which allows to decide whether a solution of the necessary first order conditions is a global minimum. This condition can be explicitly evaluated at the discrete level. Furthermore, we prove that if the above condition holds uniformly with respect to the discretization parameter the sequence of discrete solutions converges to a global solution of the corresponding limit problem. Numerical examples with unique global solutions are presented. (2000): 49J20, 35K20, 49M05, 49M25, 49M29, 65M12, 65M60 Mathematics Subject Classification

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Discrete concepts versus error analysis in PDE‐constrained optimization

Hinze

Tröltzsch

2010

GAMM-Mitteilungen

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Key words elliptic optimal control problem, state & control constraints, error analysis. MSC (2000) 49J20, 49K20, 35B37Solutions to optimization problems with pde constraints inherit special properties; the associated state solves the pde which in the optimization problem takes the role of a equality constraint, and this state together with the associated control solves an optimization problem, i.e. together with multipliers satisfies first and second order necessary optimality conditions. In this note we review the state of the art in designing discrete concepts for optimization problems with pde constraints with emphasis on structure conservation of solutions on the discrete level, and on error analysis for the discrete variables involved. As model problem for the state we consider an elliptic pde which is well understood from the analytical point of view. This allows to focus on structural aspects in discretization. We discuss the approaches First discretize, then optimize and First optimize, then discretize, and consider in detail two variants of the First discretize, then optimize approach, namely variational discretization, a discrete concept which avoids explicit discretization of the controls, and piecewise constant control approximations. We consider general constraints on the control, and also consider pointwise bounds on the state. We outline the basic ideas for providing optimal error analysis and complement our analytical findings with numerical examples which confirm our analytical results.

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Finite element approximation of elliptic control problems with constraints on the gradient

Cited by 66 publications

References 12 publications

A priori estimates for optimal Dirichlet boundary control problems

A priori estimates for optimal Dirichlet boundary control problems

Global minima for semilinear optimal control problems

Discrete concepts versus error analysis in PDE‐constrained optimization

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