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June  2021, 11(2): 313-328. doi: 10.3934/mcrf.2020038

Improved error estimates for optimal control of the Stokes problem with pointwise tracking in three dimensions

Niklas Behringer ,

Department of Mathematics, Technical University of Munich, Boltzmannstrasse 3, 85748 Garching, Germany

* Corresponding author: Niklas Behringer

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project number 188264188/GRK1754.

Received  March 2020 Revised  July 2020 Published  June 2021 Early access  August 2020

This work is motivated by recent interest in the topic of pointwise tracking type optimal control problems for the Stokes problem. Pointwise tracking consists of point evaluations in the objective functional which lead to Dirac measures appearing as source terms of the adjoint problem. Considering bounds for the control allows for improved regularity results for the exact solution and improved approximation error estimates of its numerical counterpart. We show a sub-optimal convergence result in three dimensions that nonetheless improves the results known from the literature. Finally, we offer supporting numerical experiments and insights towards optimal approximation error estimates.

Keywords: Linear-quadratic optimal control problem, Stokes equation, Dirac measures, a priori error estimates, maximum-norm estimates.
Mathematics Subject Classification: Primary:35Q35, 49M25;Secondary:35R06, 65N15, 65N30.
Citation: Niklas Behringer. Improved error estimates for optimal control of the Stokes problem with pointwise tracking in three dimensions. Mathematical Control and Related Fields, 2021, 11 (2) : 313-328. doi: 10.3934/mcrf.2020038
References:
[1]

H. W. Alt, Linear Functional Analysis, An application-oriented introduction, Translated from the German edition by Robert Nürnberg, Universitext, Springer-Verlag London, Ltd., London, 2016. doi: 10.1007/978-1-4471-7280-2.

[2]

H. Antil, E. Otárola and A. J. Salgado, Some applications of weighted norm inequalities to the error analysis of PDE-constrained optimization problems, IMA J. Numer. Anal., 38 (2018), 852-883.  doi: 10.1093/imanum/drx018.

[3]

N. Behringer, D. Leykekhman and B. Vexler., Global and local pointwise error estimates for finite element approximations to the stokes problem on convex polyhedra, SIAM J. Numer. Anal., 58(3): 1531–1555, 2020. doi: 10.1137/19M1274456.

[4]

N. Behringer, D. Meidner and B. Vexler, Finite element error estimates for optimal control problems with pointwise tracking, Pure Appl. Funct. Anal., 4 (2019), 177-204. 

[5]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, vol. 15 of Texts in Applied Mathematics, 3rd edition, Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0.

[6]

C. Brett, A. Dedner and C. Elliott, Optimal control of elliptic PDEs at points, IMA J. Numer. Anal., 36 (2016), 1015-1050.  doi: 10.1093/imanum/drv040.

[7]

R. M. Brown and Z. Shen, Estimates for the Stokes operator in Lipschitz domains, Indiana Univ. Math. J., 44 (1995), 1183-1206.  doi: 10.1512/iumj.1995.44.2025.

[8]

E. Casas, Error estimates for the numerical approximation of semilinear elliptic control problems with finitely many state constraints, A tribute to J. L. Lions, ESAIM Control Optim. Calc. Var., 8 (2002), 345-374.  doi: 10.1051/cocv:2002049.

[9]

E. Casas, M. Mateos and B. Vexler, New regularity results and improved error estimates for optimal control problems with state constraints, ESAIM Control Optim. Calc. Var., 20 (2014), 803-822.  doi: 10.1051/cocv/2013084.

[10]

L. Chang, W. Gong and N. Yan, Numerical analysis for the approximation of optimal control problems with pointwise observations, Math. Methods Appl. Sci., 38 (2015), 4502-4520.  doi: 10.1002/mma.2861.

[11]

M. Dauge, Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. I. Linearized equations, SIAM J. Math. Anal., 20 (1989), 74-97.  doi: 10.1137/0520006.

[12]

J. C. de los Reyes, C. Meyer and B. Vexler, Finite element error analysis for state-constrained optimal control of the Stokes equations, Control Cybernet., 37 (2008), 251-284. 

[13]

R. G. Durán, E. Otárola and A. J. Salgado, Stability of the Stokes projection on weighted spaces and applications, Math. Comp., 89 (2020), 1581-1603.  doi: 10.1090/mcom/3509.

[14]

F. Fuica, E. Otárola and D. Quero., Error estimates for optimal control problems involving the stokes system and dirac measures., Applied Mathematics & Optimization, Jun 2020.

[15]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Steady-State Problems. 2nd edition, Springer Monographs in Mathematics, Springer, New York, 2011. doi: 10.1007/978-0-387-09620-9.

[16]

, The finite element toolkit GASCOIGNE, http://www.gascoigne.de.

[17]

V. Girault, R. H. Nochetto and L. R. Scott, Max-norm estimates for Stokes and Navier-Stokes approximations in convex polyhedra, Numer. Math., 131 (2015), 771-822.  doi: 10.1007/s00211-015-0707-8.

[18]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, vol. 5 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.

[19]

M. Hinze, A variational discretization concept in control constrained optimization: the linear-quadratic case, Comput. Optim. Appl., 30 (2005), 45-61.  doi: 10.1007/s10589-005-4559-5.

[20]

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, vol. 31 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000, Reprint of the 1980 original. doi: 10.1137/1.9780898719451.

[21]

G. Leoni, A First Course in Sobolev Spaces, vol. 105 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2009. doi: 10.1090/gsm/105.

[22]

J.-L. Lions, Contrôle Optimal de Systèmes Gouvernés Par des Équations Aux Dérivées Partielles, Avant propos de P. Lelong, Dunod, Paris; Gauthier-Villars, Paris, 1968.

[23]

V. Maz'ya and J. Rossmann, Elliptic Equations in Polyhedral Domains, vol. 162 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/surv/162.

[24]

C. Meyer, Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints, Control Cybernet., 37 (2008), 51-83. 

[25]

RoDoBo. A C++ library for optimization with stationary and nonstationary PDEs with interface to GASCOIGNE [16], http://www.rodobo.org.

[26]

F. Tröltzsch, Optimal Control of Partial Differential Equations, Theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels, vol. 112 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/112.

[27]

W. P. Ziemer, Weakly Differentiable Functions, Sobolev spaces and functions of bounded variation, vol. 120 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.

show all references

References:
[1]

H. W. Alt, Linear Functional Analysis, An application-oriented introduction, Translated from the German edition by Robert Nürnberg, Universitext, Springer-Verlag London, Ltd., London, 2016. doi: 10.1007/978-1-4471-7280-2.

[2]

H. Antil, E. Otárola and A. J. Salgado, Some applications of weighted norm inequalities to the error analysis of PDE-constrained optimization problems, IMA J. Numer. Anal., 38 (2018), 852-883.  doi: 10.1093/imanum/drx018.

[3]

N. Behringer, D. Leykekhman and B. Vexler., Global and local pointwise error estimates for finite element approximations to the stokes problem on convex polyhedra, SIAM J. Numer. Anal., 58(3): 1531–1555, 2020. doi: 10.1137/19M1274456.

[4]

N. Behringer, D. Meidner and B. Vexler, Finite element error estimates for optimal control problems with pointwise tracking, Pure Appl. Funct. Anal., 4 (2019), 177-204. 

[5]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, vol. 15 of Texts in Applied Mathematics, 3rd edition, Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0.

[6]

C. Brett, A. Dedner and C. Elliott, Optimal control of elliptic PDEs at points, IMA J. Numer. Anal., 36 (2016), 1015-1050.  doi: 10.1093/imanum/drv040.

[7]

R. M. Brown and Z. Shen, Estimates for the Stokes operator in Lipschitz domains, Indiana Univ. Math. J., 44 (1995), 1183-1206.  doi: 10.1512/iumj.1995.44.2025.

[8]

E. Casas, Error estimates for the numerical approximation of semilinear elliptic control problems with finitely many state constraints, A tribute to J. L. Lions, ESAIM Control Optim. Calc. Var., 8 (2002), 345-374.  doi: 10.1051/cocv:2002049.

[9]

E. Casas, M. Mateos and B. Vexler, New regularity results and improved error estimates for optimal control problems with state constraints, ESAIM Control Optim. Calc. Var., 20 (2014), 803-822.  doi: 10.1051/cocv/2013084.

[10]

L. Chang, W. Gong and N. Yan, Numerical analysis for the approximation of optimal control problems with pointwise observations, Math. Methods Appl. Sci., 38 (2015), 4502-4520.  doi: 10.1002/mma.2861.

[11]

M. Dauge, Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. I. Linearized equations, SIAM J. Math. Anal., 20 (1989), 74-97.  doi: 10.1137/0520006.

[12]

J. C. de los Reyes, C. Meyer and B. Vexler, Finite element error analysis for state-constrained optimal control of the Stokes equations, Control Cybernet., 37 (2008), 251-284. 

[13]

R. G. Durán, E. Otárola and A. J. Salgado, Stability of the Stokes projection on weighted spaces and applications, Math. Comp., 89 (2020), 1581-1603.  doi: 10.1090/mcom/3509.

[14]

F. Fuica, E. Otárola and D. Quero., Error estimates for optimal control problems involving the stokes system and dirac measures., Applied Mathematics & Optimization, Jun 2020.

[15]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Steady-State Problems. 2nd edition, Springer Monographs in Mathematics, Springer, New York, 2011. doi: 10.1007/978-0-387-09620-9.

[16]

, The finite element toolkit GASCOIGNE, http://www.gascoigne.de.

[17]

V. Girault, R. H. Nochetto and L. R. Scott, Max-norm estimates for Stokes and Navier-Stokes approximations in convex polyhedra, Numer. Math., 131 (2015), 771-822.  doi: 10.1007/s00211-015-0707-8.

[18]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, vol. 5 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.

[19]

M. Hinze, A variational discretization concept in control constrained optimization: the linear-quadratic case, Comput. Optim. Appl., 30 (2005), 45-61.  doi: 10.1007/s10589-005-4559-5.

[20]

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, vol. 31 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000, Reprint of the 1980 original. doi: 10.1137/1.9780898719451.

[21]

G. Leoni, A First Course in Sobolev Spaces, vol. 105 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2009. doi: 10.1090/gsm/105.

[22]

J.-L. Lions, Contrôle Optimal de Systèmes Gouvernés Par des Équations Aux Dérivées Partielles, Avant propos de P. Lelong, Dunod, Paris; Gauthier-Villars, Paris, 1968.

[23]

V. Maz'ya and J. Rossmann, Elliptic Equations in Polyhedral Domains, vol. 162 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/surv/162.

[24]

C. Meyer, Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints, Control Cybernet., 37 (2008), 51-83. 

[25]

RoDoBo. A C++ library for optimization with stationary and nonstationary PDEs with interface to GASCOIGNE [16], http://www.rodobo.org.

[26]

F. Tröltzsch, Optimal Control of Partial Differential Equations, Theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels, vol. 112 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/112.

[27]

W. P. Ziemer, Weakly Differentiable Functions, Sobolev spaces and functions of bounded variation, vol. 120 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.

Figure 1.  Threshold visualization of the first component of a solution $ \vec{q}_h $ to Problem (16)
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Figure 2.  Error $ ||\bar q_{n}-\bar q_h||_{L^2(\Omega)} $ for cellwise constant control discretization and different choices for the bounds $ \vec a $ and $ \vec b $. $ \bar q_{n} $ denotes the approximate solution on a finer mesh
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