doi: 10.3934/mcrf.2020038

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

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  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.

Citation: Niklas Behringer. Improved error estimates for optimal control of the Stokes problem with pointwise tracking in three dimensions. Mathematical Control & Related Fields, 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.  Google Scholar

[2]

H. AntilE. 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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

E. CasasM. 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.  Google Scholar

[10]

L. ChangW. 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.  Google Scholar

[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.  Google Scholar

[12]

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

[13]

R. G. DuránE. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[16]

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

[17]

V. GiraultR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

H. AntilE. 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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

E. CasasM. 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.  Google Scholar

[10]

L. ChangW. 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.  Google Scholar

[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.  Google Scholar

[12]

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

[13]

R. G. DuránE. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[16]

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

[17]

V. GiraultR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Threshold visualization of the first component of a solution $ \vec{q}_h $ to Problem (16)
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
[1]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[2]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[3]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168

[4]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048

[5]

Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019

[6]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377

[7]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[8]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046

[9]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[10]

Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168

[11]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[12]

Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016

[13]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[14]

Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347

[15]

Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032

[16]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

[17]

Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297

[18]

Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020374

[19]

Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169

[20]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241

2019 Impact Factor: 0.857

Article outline

Figures and Tables

[Back to Top]