A Priori Error Analysis of the Petrov–Galerkin Crank–Nicolson Scheme for Parabolic Optimal Control Problems
Abstract
In this paper, a finite element discretization of an optimal control problem governed by the heat equation is considered. The temporal discretization is based on a Petrov–Galerkin variant of the Crank–Nicolson scheme, whereas the spatial discretization employs usual conforming finite elements. With a suitable postprocessing step, a discrete solution is obtained for which error estimates of optimal order are proven. A numerical result is presented for illustrating the theoretical findings.
MSC codes
Keywords
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
T. Apel and T. Flaig, Crank-Nicolson Schemes for Optimal Control Problems with Evolution Equations, Preprint SPP1253–113, DFG priority program 1253 “Optimization with PDEs,” Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Germany, 2010.
2.
A. Aziz and P. Monk, Continuous finite elements in space and time for the heat equation, Math. Comp., 52 (1989), pp. 255–274.
3.
R. Becker, D. Meidner, and B. Vexler, Efficient numerical solution of parabolic optimization problems by finite element methods, Optim. Methods Softw., 22 (2007), pp. 813–833.
4.
R. Becker and B. Vexler, Optimal control of the convection-diffusion equation using stabilized finite element methods, Numer. Math., 106 (2007), pp. 349–367.
5.
M. Bergounioux, K. Ito, and K. Kunisch, Primal-dual strategy for constrained optimal control problems, SIAM J. Control Optim., 37 (1999), pp. 1176–1194.
6.
E. Burman, Crank-Nicolson finite element methods using symmetric stabilization with an application to optimal control problems subject to transient advection-diffusion equations, Commun. Math. Sci., 9 (2011), pp. 319–329.
7.
K. Chrysafinos, Convergence of discontinuous Galerkin approximations of an optimal control problem associated to semilinear parabolic PDE's, M2AN Math. Model. Numer. Anal., 44 (2010), pp. 189–206.
8.
P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Classics Appl. Math. 40, SIAM, Philadelphia, 2002.
9.
R. Dautray and J.-L. Lions, Evolution Problems I, Math. Anal. Numer. Methods Sci. Technol. 5, Springer-Verlag, Berlin, 1992.
10.
K. Deckelnick and M. Hinze, Variational discretization of parabolic control problems in the presence of pointwise state constraints, J. Comput. Math., 29 (2011), pp. 1–16.
11.
K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Computational Differential Equations, Cambridge University Press, Cambridge, UK, 1996.
12.
L. C. Evans, Partial Differential Equations, Grad. Stud. Math. 19, AMS, Providence, RI, 2002.
13.
Gascoigne, High Performance Finite Element Toolkit, http://www.gascoigne.uni-hd.de (2002).
14.
K. Kunisch and A. Rösch, Primal-dual active set strategy for a general class of constrained optimal control problems, SIAM J. Optim., 13 (2002), pp. 321–334.
15.
I. Lasiecka and K. Malanowski, On discrete-time Ritz-Galerkin approximation of control constrained optimal control problems for parabolic systems, Control Cybernet., 7 (1978), pp. 21–36.
16.
J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Grundlehren Math. Wiss. 170, Springer, Berlin, 1971.
17.
M. Luskin and R. Rannacher, On the smoothing property of the Galerkin method for parabolic equations, SIAM J. Numer. Anal., 19 (1982), pp. 93–113.
18.
K. Malanowski, Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal-control problems, Appl. Math. Optim., 8 (1981), pp. 69–95.
19.
R. S. McNight and W. E. Bosarge Jr., The Ritz–Galerkin procedure for parabolic control problems, SIAM J. Control Optim., 11 (1973), pp. 510–524.
20.
D. Meidner, R. Rannacher, and B. Vexler, A priori error estimates for finite element discretizations of parabolic optimization problems with pointwise state constraints in time, SIAM J. Control Optim., 49 (2011), pp. 1961–1997.
21.
D. Meidner and B. Vexler, Adaptive space-time finite element methods for parabolic optimization problems, SIAM J. Control Optim., 46 (2007), pp. 116–142.
22.
D. Meidner and B. Vexler, A priori error estimates for space-time finite element discretization of parabolic optimal control problems. Part I: Problems without control constraints, SIAM J. Control Optim., 47 (2008), pp. 1150–1177.
23.
D. Meidner and B. Vexler, A priori error estimates for space-time finite element discretization of parabolic optimal control problems. Part II: Problems with control constraints, SIAM J. Control Optim., 47 (2008), pp. 1301–1329.
24.
C. Meyer and A. Rösch, Superconvergence properties of optimal control problems, SIAM J. Control Optim., 43 (2004), pp. 970–985.
25.
I. Neitzel and B. Vexler, A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems, Numer. Math., to appear; published online September 15, 2011 (
26.
R. Rannacher, Finite element solution of diffusion problems with irregular data, Numer. Math., 43 (1984), pp. 309–327.
27.
RoDoBo, A C$++$ Library for Optimization with Stationary and Nonstationary PDEs with interface to Gascoigne, http://www.rodobo.uni-hd.de (2010).
28.
A. Rösch, Error estimates for parabolic optimal control problems with control constraints, Z. Anal. Anwendungen, 23 (2004), pp. 353–376.
29.
A. Rösch and B. Vexler, Optimal control of the Stokes equations: A priori error analysis for finite element discretization with postprocessing, SIAM J. Numer. Anal., 44 (2006), pp. 1903–1920.
30.
F. Schieweck, A-stable discontinuous Galerkin-Petrov time discretization of higher order, J. Numer. Math., 18 (2010), p. 25–57.
31.
V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer Ser. Comput. Math. 25, Springer, Berlin, 2006.
32.
R. Winther, Error estimates for a Galerkin approximation of a parabolic control problem, Ann. Mat. Pura Appl. (4), 117 (1978), pp. 173–206.
33.
J. Wloka, Partielle Differentialgleichungen: Sobolevräume und Randwertaufgaben, B. G. Teubner, Stuttgart, 1982.
Information & Authors
Information
Published In
SIAM Journal on Control and Optimization
Pages: 2183 - 2211
DOI: 10.1137/100809611
ISSN (online): 1095-7138
Copyright
Copyright © 2011 Society for Industrial and Applied Mathematics.
History
Submitted: 23 September 2010
Accepted: 2 August 2011
Published online: 27 October 2011
MSC codes
Keywords
Authors
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- An adaptive certified space-time reduced basis method for nonsmooth parabolic partial differential equationsAdvances in Computational Mathematics, Vol. 50, No. 3 | 15 May 2024
- A space–time variational method for optimal control problems: well-posedness, stability and numerical solutionComputational Optimization and Applications, Vol. 86, No. 2 | 25 July 2023
- Implicit Runge–Kutta Schemes for Optimal Control Problems with Evolution EquationsComputational Methods in Applied Mathematics, Vol. 23, No. 4 | 1 April 2023
- A priori error estimates of Crank-Nicolson finite element method for parabolic optimal control problemsComputers & Mathematics with Applications, Vol. 144 | 1 Aug 2023
- Error estimates for the numerical approximation of optimal control problems with nonsmooth pointwise-integral control constraintsIMA Journal of Numerical Analysis, Vol. 43, No. 3 | 7 July 2022
- A two-layer Crank–Nicolson linear finite element methods for second-order hyperbolic optimal control problemsResults in Applied Mathematics, Vol. 18 | 1 May 2023
- Error estimates of Fourier finite volume element method for parabolic Dirichlet boundary optimal control problems on complex connected domainsApplied Numerical Mathematics, Vol. 186 | 1 Apr 2023
- Error estimates of mixed finite elements combined with Crank-Nicolson scheme for parabolic control problemsAIMS Mathematics, Vol. 8, No. 5 | 1 Jan 2023
- Continuous Galerkin schemes for semiexplicit differential-algebraic equationsIMA Journal of Numerical Analysis, Vol. 42, No. 3 | 18 May 2021
- Mixed and hybrid Petrov–Galerkin finite element discretization for optimal control of the wave equationNumerische Mathematik, Vol. 150, No. 2 | 12 January 2022
- Temporally Semidiscrete Approximation of a Dirichlet Boundary Control for a Fractional/Normal Evolution Equation with a Final ObservationJournal of Scientific Computing, Vol. 88, No. 1 | 21 May 2021
- Numerical Approximation of Quadratic OCPs Governed by Linear Evolution PDEsOptimal Control of Partial Differential Equations | 1 January 2022
- Adaptive variational discretization approximation method for parabolic optimal control problemsJournal of Inequalities and Applications, Vol. 2020, No. 1 | 11 November 2020
- Second Order Pressure Estimates for the Crank--Nicolson Discretization of the Incompressible Navier--Stokes EquationsSIAM Journal on Numerical Analysis, Vol. 58, No. 1 | 16 January 2020
- Immersed finite element method and its analysis for parabolic optimal control problems with interfacesApplied Numerical Mathematics, Vol. 147 | 1 Jan 2020
- Optimal control problem of the two-dimensional modified anomalous subdiffusion equation with discontinuous Galerkin approximationComputers & Mathematics with Applications, Vol. 78, No. 6 | 1 Sep 2019
- Optimal time delays in a class of reaction-diffusion equationsOptimization, Vol. 68, No. 1 | 15 August 2018
- Error estimates of the space–time spectral method for parabolic control problemsComputers & Mathematics with Applications, Vol. 75, No. 2 | 1 Jan 2018
- A new multigrid method for unconstrained parabolic optimal control problemsJournal of Computational and Applied Mathematics, Vol. 326 | 1 Dec 2017
- Discrete maximal parabolic regularity for Galerkin finite element methodsNumerische Mathematik, Vol. 135, No. 3 | 30 June 2016
- Optimal Control of a Linear Unsteady Fluid–Structure Interaction ProblemJournal of Optimization Theory and Applications, Vol. 170, No. 1 | 4 April 2016
- Finite Element Method and A Priori Error Estimates for Dirichlet Boundary Control Problems Governed by Parabolic PDEsJournal of Scientific Computing, Vol. 66, No. 3 | 4 June 2015
- A Review of Numerical Analysis for the Discretization of the Velocity Tracking ProblemTrends in Differential Equations and Applications | 10 June 2016
- A leapfrog semi-smooth Newton-multigrid method for semilinear parabolic optimal control problemsComputational Optimization and Applications, Vol. 63, No. 1 | 12 May 2015
- Error estimates for the discretization of the velocity tracking problemNumerische Mathematik, Vol. 130, No. 4 | 23 November 2014
- A posteriori error estimation for the fractional step theta discretization of the incompressible Navier–Stokes equationsComputer Methods in Applied Mechanics and Engineering, Vol. 288 | 1 May 2015
- Crank--Nicolson Time Stepping and Variational Discretization of Control-Constrained Parabolic Optimal Control ProblemsSIAM Journal on Control and Optimization, Vol. 53, No. 3 | 7 May 2015
- A priori error analysis of the upwind symmetric interior penalty Galerkin (SIPG) method for the optimal control problems governed by unsteady convection diffusion equationsComputational Optimization and Applications, Vol. 57, No. 3 | 25 September 2013
- A Priori Error Estimates for Finite Element Methods for H(2,1) -Elliptic EquationsNumerical Functional Analysis and Optimization, Vol. 35, No. 2 | 30 December 2013
- Petrov-Galerkin Crank-Nicolson Scheme for Parabolic Optimal Control Problems on Nonsmooth DomainsTrends in PDE Constrained Optimization | 22 November 2014
- Third order convergent time discretization for parabolic optimal control problems with control constraintsComputational Optimization and Applications, Vol. 57, No. 1 | 17 July 2013
- Efficient numerical realization of discontinuous Galerkin methods for temporal discretization of parabolic problemsNumerische Mathematik, Vol. 124, No. 1 | 23 October 2012
- Crank--Nicolson Schemes for Optimal Control Problems with Evolution EquationsSIAM Journal on Numerical Analysis, Vol. 50, No. 3 | 5 June 2012
- A Discontinuous Galerkin Time-Stepping Scheme for the Velocity Tracking ProblemSIAM Journal on Numerical Analysis, Vol. 50, No. 5 | 18 September 2012
- Symmetric error estimates for discontinuous Galerkin approximations for an optimal control problem associated to semilinear parabolic PDE'sDiscrete & Continuous Dynamical Systems - B, Vol. 17, No. 5 | 1 Jan 2012
View Options
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].