Convergence of discontinuous Galerkin approximations of an optimal control problem associated to semilinear parabolic PDE's

Chrysafinos, Konstantinos

doi:10.1051/m2an/2009046

Cited by 18 publications

(12 citation statements)

References 43 publications

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“…Only few other results have been published on error estimates for parabolic optimization problems. We mention the papers [18,19,26,29], which are concerned with linear-quadratic problems and a recent article [6], where pure convergence (without rates) of discontinuous Galerkin schemes for control problems governed by semilinear parabolic equations has been shown. On the contrary, quite a number of results are known for elliptic problems, cf., e.g., [1,4,13,[15][16][17]24].…”

Section: Introductionmentioning

confidence: 99%

A priori error estimates for space–time finite element discretization of semilinear parabolic optimal control problems

Neitzel

Vexler

2011

Numer. Math.

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In this paper, a priori error estimates for space-time finite element discretizations of optimal control problems governed by semilinear parabolic PDEs and subject to pointwise control constraints are derived. We extend the approach from Meidner and Vexler (SIAM Control Optim 47(3):1150-1177, 2008; SIAM Control Optim 47 (3):1301-1329, 2008) where linear-quadratic problems have been considered, discretizing the state equation by usual conforming finite elements in space and a discontinuous Galerkin method in time. Error estimates for controls discretized by piecewise constant functions in time and cellwise constant functions in space are derived in detail and we explain how error estimate for further discretization approaches, e.g., cellwise linear discretization in space, the postprocessing approach from Meyer and Rösch (SIAM J Control Optim 43: [970][971][972][973][974][975][976][977][978][979][980][981][982][983][984][985] 2004), and the variationally discrete approach from Hinze (J Comput Optim Appl 30:45-63, 2005) can be obtained. In addition, we derive an estimate for a setting with finitely many timedependent controls. Mathematics Subject Classification (2000)49M25 · 65M15 · 65M60 Supported by DFG priority program SPP1253. I. Neitzel

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Section: Introductionmentioning

confidence: 99%

A priori error estimates for space–time finite element discretization of semilinear parabolic optimal control problems

Neitzel

Vexler

2011

Numer. Math.

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“…Moreover, the discretization in time of the state equation leads to a stationary Navier-Stokes system, for which we cannot guarantee the uniqueness of a solution. Finally in [19] a convergence result for an optimal control problem related to semi-linear parabolic pdes is presented under minimal regularity assumptions on the given data.…”

Section: Related Resultsmentioning

confidence: 99%

“…Error estimates for discontinuous time-stepping schemes for distributed optimal control problems related to linear parabolic pdes with possibly time dependent coefficients, were presented in [17,18]. An analysis of second order Petrov-Galerkin Crank-Nicolson scheme and of a Crank-Nicolson scheme, for an optimal control problem for the heat equation were analyzed in [2,39] respectively where estimates of second-order (in time) are derived.…”

Section: Related Resultsmentioning

confidence: 99%

A Review of Numerical Analysis for the Discretization of the Velocity Tracking Problem

Casas

Chrysafinos

2016

SEMA SIMAI Springer Series

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In this paper we are reviewing results regarding the velocity tracking problem. In particular, we focus on our work (Casas and Chrysafinos, SIAM J. Numer. Anal. 50(5): 2281-2306, 2012 Casas and Chrysafinos, Numer. Math. 130:615-643, 2015; and Casas and Crysafinos, to appear in ESAIM: COCV) concerning a-priori error estimates for the velocity tracking of two-dimensional evolutionary Navier-Stokes flows. The controls are of distributed type, and subject to point-wise control constraints. The standard tracking type functional is considered, however the option of setting the penalty-regularization parameter D 0 in front of the L 2 .0; TI L 2 .˝// norm of the control in the functional is also discussed. The discretization scheme of the state and adjoint equations is based on a discontinuous time-stepping scheme combined with conforming finite elements (in space) for the velocity and pressure. Provided that the time and space discretization parameters, and h respectively, satisfy Ä p / for some p > 2, are discussed for the difference between the locally optimal controls and their discrete approximations, when the controls are discretized by piecewise constants functions, the variational discretization approach or by using piecewise-linears in space respectively for > 0. For the case of D 0, (bangbang type controls) we also discuss various issues related to the analysis and discretization, emphasizing on the different features compared to the case > 0. In particular, fully-discrete estimates for the states are presented and discussed.

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“…For earlier work on these schemes within the context of optimal control problems we refer the reader to [32], [33] for error estimates for an optimal control problem for the heat equation, with and without control constraints respectively, and to [12] for a convergence result for a semilinear parabolic optimal control problem. Error estimates for higher order discontinuous time stepping schemes were presented in [11], while an analysis of second order Petrov-Galerkin Crank-Nicolson scheme and of a Crank-Nicolson scheme, for an optimal control problem for the heat equation were analyzed in [34] and [2] respectively where estimates of second-order (in time) are de-rived.…”

Section: Related Resultsmentioning

confidence: 99%

Error estimates for the discretization of the velocity tracking problem

Casas

Chrysafinos

2014

Numer. Math.

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Summary In this paper we are continuing our work [6], concerning a-priori error estimates for the velocity tracking of two-dimensional evolutionary Navier-Stokes flows. The controls are of distributed type, and subject to point-wise control constraints. The discretization scheme of the state and adjoint equations is based on a discontinuous timestepping scheme (in time) combined with conforming finite elements (in space) for the velocity and pressure. Provided that the time and space discretization parameters, τ and h respectively, satisfy τ ≤ Ch 2 , error estimates of order O(h 2 ) and O(h 3 2 − 2 p ) with p > 3 depending on the regularity of the target and the initial velocity, are proved for the difference between the locally optimal controls and their discrete approximations, when the controls are discretized by the variational discretization approach and by using piecewise-linear functions in space respectively. Both results are based on new duality arguments for the evolutionary Navier-Stokes equations.

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Convergence of discontinuous Galerkin approximations of an optimal control problem associated to semilinear parabolic PDE's

Cited by 18 publications

References 43 publications

A priori error estimates for space–time finite element discretization of semilinear parabolic optimal control problems

A priori error estimates for space–time finite element discretization of semilinear parabolic optimal control problems

A Review of Numerical Analysis for the Discretization of the Velocity Tracking Problem

Error estimates for the discretization of the velocity tracking problem

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