Figure 6.
Constrained problems. Experimental orders of convergence vs biggest angle. Left: generic case. Right: worst case.
-
Previous Article
Error analysis for global minima of semilinear optimal control problems
- MCRF Home
- This Issue
-
Next Article
Optimal control of a non-smooth semilinear elliptic equation
March
2018, 8(1): 217-245.
doi: 10.3934/mcrf.2018010
Error estimates for Dirichlet control problems in polygonal domains: Quasi-uniform meshes
| 1. | Institut für Mathematik und Bauinformatik, Universität der Bundeswehr München, 85577 Neubiberg, Germany |
| 2. | Departamento de Matemáticas, Universidad de Oviedo, 33203 Gijón, Spain |
| 3. | Lehrstuhl für Optimalsteuerung, Technische Universität München, 85748 Garching bei München, Germany |
| 4. | Fakultät für Mathematik, Universtät Duisburg-Essen, D-45127 Essen, Germany |
The paper deals with finite element approximations of elliptic Dirichlet boundary control problems posed on two-dimensional polygonal domains. Error estimates are derived for the approximation of the control and the state variables. Special features of unconstrained and control constrained problems as well as general quasi-uniform meshes and superconvergent meshes are carefully elaborated. Compared to existing results, the convergence rates for the control variable are not only improved but also fully explain the observed orders of convergence in the literature. Moreover, for the first time, results in nonconvex domains are provided.
Keywords:
Optimal control,
boundary control,
Dirichlet control,
nonconvex domain,
finite elements,
error estimates,
superconvergent meshes.
Mathematics Subject Classification:
65N30, 65N15, 49M05, 49M25.
Citation:
Thomas Apel, Mariano Mateos, Johannes Pfefferer, Arnd Rösch. Error estimates for Dirichlet control problems in polygonal domains: Quasi-uniform meshes. Mathematical Control & Related Fields,
2018, 8
(1)
: 217-245.
doi: 10.3934/mcrf.2018010
![]()
References:
| [1] |
T. Apel, M. Mateos, J. Pfefferer and A. Rösch,
On the regularity of the solutions of Dirichlet optimal control problems in polygonal domains, SIAM J. Control Optim., 53 (2015), 3620-3641.
doi: 10.1137/140994186. |
| [2] |
T. Apel, S. Nicaise and J. Pfefferer,
Discretization of the Poisson equation with non-smooth data and emphasis on non-convex domains, Numer. Methods Partial Differential Equations, 32 (2016), 1433-1454.
doi: 10.1002/num.22057. |
| [3] |
T. Apel, J. Pfefferer and A. Rösch,
Finite element error estimates on the boundary with application to optimal control, Mathematics of Computation, 84 (2015), 33-70.
doi: 10.1090/S0025-5718-2014-02862-7. |
| [4] |
C. Bacuta, J. Bramble and J. Xu,
Regularity estimates for elliptic boundary value problems in Besov spaces, Mathematics of Computation, 72 (2003), 1577-1595.
doi: 10.1090/S0025-5718-02-01502-8. |
| [5] |
R. Bank and J. Xu,
Asymptotically exact a posteriori error estimators, part Ⅰ: Grids with superconvergence, SIAM Journal on Numerical Analysis, 41 (2003), 2294-2312.
doi: 10.1137/S003614290139874X. |
| [6] |
S. Bartels, C. Carstensen and G. Dolzmann,
Inhomogeneous Dirichlet conditions in a priori and a posteriori finite element error analysis, Numerische Mathematik, 99 (2004), 1-24.
doi: 10.1007/s00211-004-0548-3. |
| [7] |
M. Berggren,
Approximations of very weak solutions to boundary-value problems, SIAM J. Numer. Anal., 42 (2004), 860-877 (electronic).
doi: 10.1137/S0036142903382048. |
| [8] |
E. Casas and J.-P. Raymond,
Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations, SIAM J. Control Optim., 45 (2006), 1586-1611 (electronic).
doi: 10.1137/050626600. |
| [9] |
E. Casas and F. Tröltzsch, Error estimates for linear-quadratic elliptic control problems, in Analysis and Optimization of Differential Systems, Springer, 2003, 89-100. |
| [10] |
P. Ciarlet, Basic error estimates for elliptic problems, in Handbook of Numerical Analysis (eds. P. Ciarlet and J. Lions), vol. Ⅱ. Finite Element Methods (Part 1), North-Holland, 1991, 17-352. |
| [11] |
M. Costabel,
Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal., 19 (1988), 613-626.
doi: 10.1137/0519043. |
| [12] |
K. Deckelnick, A. Günther and M. Hinze,
Finite element approximation of Dirichlet boundary control for elliptic PDEs on two-and three-dimensional curved domains, SIAM J. Control Optim., 48 (2009), 2798-2819.
doi: 10.1137/080735369. |
| [13] |
A. Demlow, D. Leykekhman, A. Schatz and L. Wahlbin,
Best approximation property in the w∞1 norm for finite element methods on graded meshes, Mathematics of Computation, 81 (2012), 743-764.
doi: 10.1090/S0025-5718-2011-02546-9. |
| [14] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985. |
| [15] |
T. Horger, J. Melenk and B. Wohlmuth,
On optimal L2-and surface flux convergence in FEM, Computing and Visualization in Science, 16 (2013), 231-246.
doi: 10.1007/s00791-015-0237-z. |
| [16] |
M. Mateos, Optimization methods for Dirichlet control problems, to appear in Optimization, https://arXiv.org/abs/1701.07619. |
| [17] |
M. Mateos and I. Neitzel,
Dirichlet control of elliptic state constrained problems, Comput. Optim. Appl., 63 (2016), 825-853.
doi: 10.1007/s10589-015-9784-y. |
| [18] |
S. May, R. Rannacher and B. Vexler,
Error analysis for a finite element approximation of elliptic Dirichlet boundary control problems, SIAM Journal on Control and Optimization, 51 (2013), 2585-2611.
doi: 10.1137/080735734. |
| [19] |
J. Melenk and B. Wohlmuth,
Quasi-optimal approximation of surface based lagrange multipliers in finite element methods, SIAM Journal on Numerical Analysis, 50 (2012), 2064-2087.
doi: 10.1137/110832999. |
| [20] |
S. Nazarov and B. A. Plamenevsky, Elliptic Problems in Domains with Piecewise Smooth Boundaries, vol. 13 of De Gruyter Expositions in Mathematics, Walter de Gruyter & Co., Berlin, 1994.
doi: 10.1515/9783110848915.525. |
| [21] |
J. Nečas, Direct Methods in the Theory of Elliptic Equations, Corrected 2nd edition, Monographs and Studies in Mathematics, Springer Berlin Heidelberg, 2012. |
| [22] |
J. Pfefferer, Numerical analysis for elliptic Neumann boundary control problems on polygonal domains, PhD Thesis, Universität der Bundeswehr München, 2014, http://athene.bibl.unibw-muenchen.de:8081/node?id=92055. |
| [23] |
R. Rannacher and R. Scott,
Some optimal error estimates for piecewise linear finite element approximations, Mathematics of Computation, 38 (1982), 437-445.
doi: 10.1090/S0025-5718-1982-0645661-4. |
show all references
References:
| [1] |
T. Apel, M. Mateos, J. Pfefferer and A. Rösch,
On the regularity of the solutions of Dirichlet optimal control problems in polygonal domains, SIAM J. Control Optim., 53 (2015), 3620-3641.
doi: 10.1137/140994186. |
| [2] |
T. Apel, S. Nicaise and J. Pfefferer,
Discretization of the Poisson equation with non-smooth data and emphasis on non-convex domains, Numer. Methods Partial Differential Equations, 32 (2016), 1433-1454.
doi: 10.1002/num.22057. |
| [3] |
T. Apel, J. Pfefferer and A. Rösch,
Finite element error estimates on the boundary with application to optimal control, Mathematics of Computation, 84 (2015), 33-70.
doi: 10.1090/S0025-5718-2014-02862-7. |
| [4] |
C. Bacuta, J. Bramble and J. Xu,
Regularity estimates for elliptic boundary value problems in Besov spaces, Mathematics of Computation, 72 (2003), 1577-1595.
doi: 10.1090/S0025-5718-02-01502-8. |
| [5] |
R. Bank and J. Xu,
Asymptotically exact a posteriori error estimators, part Ⅰ: Grids with superconvergence, SIAM Journal on Numerical Analysis, 41 (2003), 2294-2312.
doi: 10.1137/S003614290139874X. |
| [6] |
S. Bartels, C. Carstensen and G. Dolzmann,
Inhomogeneous Dirichlet conditions in a priori and a posteriori finite element error analysis, Numerische Mathematik, 99 (2004), 1-24.
doi: 10.1007/s00211-004-0548-3. |
| [7] |
M. Berggren,
Approximations of very weak solutions to boundary-value problems, SIAM J. Numer. Anal., 42 (2004), 860-877 (electronic).
doi: 10.1137/S0036142903382048. |
| [8] |
E. Casas and J.-P. Raymond,
Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations, SIAM J. Control Optim., 45 (2006), 1586-1611 (electronic).
doi: 10.1137/050626600. |
| [9] |
E. Casas and F. Tröltzsch, Error estimates for linear-quadratic elliptic control problems, in Analysis and Optimization of Differential Systems, Springer, 2003, 89-100. |
| [10] |
P. Ciarlet, Basic error estimates for elliptic problems, in Handbook of Numerical Analysis (eds. P. Ciarlet and J. Lions), vol. Ⅱ. Finite Element Methods (Part 1), North-Holland, 1991, 17-352. |
| [11] |
M. Costabel,
Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal., 19 (1988), 613-626.
doi: 10.1137/0519043. |
| [12] |
K. Deckelnick, A. Günther and M. Hinze,
Finite element approximation of Dirichlet boundary control for elliptic PDEs on two-and three-dimensional curved domains, SIAM J. Control Optim., 48 (2009), 2798-2819.
doi: 10.1137/080735369. |
| [13] |
A. Demlow, D. Leykekhman, A. Schatz and L. Wahlbin,
Best approximation property in the w∞1 norm for finite element methods on graded meshes, Mathematics of Computation, 81 (2012), 743-764.
doi: 10.1090/S0025-5718-2011-02546-9. |
| [14] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985. |
| [15] |
T. Horger, J. Melenk and B. Wohlmuth,
On optimal L2-and surface flux convergence in FEM, Computing and Visualization in Science, 16 (2013), 231-246.
doi: 10.1007/s00791-015-0237-z. |
| [16] |
M. Mateos, Optimization methods for Dirichlet control problems, to appear in Optimization, https://arXiv.org/abs/1701.07619. |
| [17] |
M. Mateos and I. Neitzel,
Dirichlet control of elliptic state constrained problems, Comput. Optim. Appl., 63 (2016), 825-853.
doi: 10.1007/s10589-015-9784-y. |
| [18] |
S. May, R. Rannacher and B. Vexler,
Error analysis for a finite element approximation of elliptic Dirichlet boundary control problems, SIAM Journal on Control and Optimization, 51 (2013), 2585-2611.
doi: 10.1137/080735734. |
| [19] |
J. Melenk and B. Wohlmuth,
Quasi-optimal approximation of surface based lagrange multipliers in finite element methods, SIAM Journal on Numerical Analysis, 50 (2012), 2064-2087.
doi: 10.1137/110832999. |
| [20] |
S. Nazarov and B. A. Plamenevsky, Elliptic Problems in Domains with Piecewise Smooth Boundaries, vol. 13 of De Gruyter Expositions in Mathematics, Walter de Gruyter & Co., Berlin, 1994.
doi: 10.1515/9783110848915.525. |
| [21] |
J. Nečas, Direct Methods in the Theory of Elliptic Equations, Corrected 2nd edition, Monographs and Studies in Mathematics, Springer Berlin Heidelberg, 2012. |
| [22] |
J. Pfefferer, Numerical analysis for elliptic Neumann boundary control problems on polygonal domains, PhD Thesis, Universität der Bundeswehr München, 2014, http://athene.bibl.unibw-muenchen.de:8081/node?id=92055. |
| [23] |
R. Rannacher and R. Scott,
Some optimal error estimates for piecewise linear finite element approximations, Mathematics of Computation, 38 (1982), 437-445.
doi: 10.1090/S0025-5718-1982-0645661-4. |
Figure 1.
Convergence rates depending on the maximal interior angle in the unconstrained case
Figure 2.
Convergence rates depending on the maximal interior angle in the constrained case
Figure 3.
Family of quasi-uniform meshes which is not $O(h^2)$ -irregular
Figure 4.
Family of quasi-uniform $O(h^2)$ -irregular meshes
Figure 5.
Unconstrained problems. Experimental orders of convergence vs biggest angle.
| [1] |
Tianliang Hou, Yanping Chen. Superconvergence for elliptic optimal control problems discretized by RT1 mixed finite elements and linear discontinuous elements. Journal of Industrial & Management Optimization, 2013, 9 (3) : 631-642. doi: 10.3934/jimo.2013.9.631 |
| [2] |
Shaolin Ji, Xiaole Xue. A stochastic maximum principle for linear quadratic problem with nonconvex control domain. Mathematical Control & Related Fields, 2019, 9 (3) : 495-507. doi: 10.3934/mcrf.2019022 |
| [3] |
Konstantinos Chrysafinos, Efthimios N. Karatzas. Symmetric error estimates for discontinuous Galerkin approximations for an optimal control problem associated to semilinear parabolic PDE's. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1473-1506. doi: 10.3934/dcdsb.2012.17.1473 |
| [4] |
Luca Di Persio, Giacomo Ziglio. Gaussian estimates on networks with applications to optimal control. Networks & Heterogeneous Media, 2011, 6 (2) : 279-296. doi: 10.3934/nhm.2011.6.279 |
| [5] |
Simone Göttlich, Patrick Schindler. Optimal inflow control of production systems with finite buffers. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 107-127. doi: 10.3934/dcdsb.2015.20.107 |
| [6] |
Philip Trautmann, Boris Vexler, Alexander Zlotnik. Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients. Mathematical Control & Related Fields, 2018, 8 (2) : 411-449. doi: 10.3934/mcrf.2018017 |
| [7] |
Ahmad Ahmad Ali, Klaus Deckelnick, Michael Hinze. Error analysis for global minima of semilinear optimal control problems. Mathematical Control & Related Fields, 2018, 8 (1) : 195-215. doi: 10.3934/mcrf.2018009 |
| [8] |
Alexander Arguchintsev, Vasilisa Poplevko. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 193-202. doi: 10.3934/naco.2018011 |
| [9] |
Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025 |
| [10] |
Ciro D’Apice, Umberto De Maio, Peter I. Kogut. Boundary velocity suboptimal control of incompressible flow in cylindrically perforated domain. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 283-314. doi: 10.3934/dcdsb.2009.11.283 |
| [11] |
Tan H. Cao, Boris S. Mordukhovich. Applications of optimal control of a nonconvex sweeping process to optimization of the planar crowd motion model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-26. doi: 10.3934/dcdsb.2019078 |
| [12] |
Fabio Bagagiolo. Optimal control of finite horizon type for a multidimensional delayed switching system. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 239-264. doi: 10.3934/dcdsb.2005.5.239 |
| [13] |
Chunjuan Hou, Yanping Chen, Zuliang Lu. Superconvergence property of finite element methods for parabolic optimal control problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 927-945. doi: 10.3934/jimo.2011.7.927 |
| [14] |
Gang Chen, Zaiming Liu, Jinbiao Wu. Optimal threshold control of a retrial queueing system with finite buffer. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1537-1552. doi: 10.3934/jimo.2017006 |
| [15] |
Divya Thakur, Belinda Marchand. Hybrid optimal control for HIV multi-drug therapies: A finite set control transcription approach. Mathematical Biosciences & Engineering, 2012, 9 (4) : 899-914. doi: 10.3934/mbe.2012.9.899 |
| [16] |
Thalya Burden, Jon Ernstberger, K. Renee Fister. Optimal control applied to immunotherapy. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 135-146. doi: 10.3934/dcdsb.2004.4.135 |
| [17] |
Ellina Grigorieva, Evgenii Khailov. Optimal control of pollution stock. Conference Publications, 2011, 2011 (Special) : 578-588. doi: 10.3934/proc.2011.2011.578 |
| [18] |
Hang-Chin Lai, Jin-Chirng Lee, Shuh-Jye Chern. A variational problem and optimal control. Journal of Industrial & Management Optimization, 2011, 7 (4) : 967-975. doi: 10.3934/jimo.2011.7.967 |
| [19] |
Orazio Arena. On some boundary control problems. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 613-618. doi: 10.3934/dcdss.2016015 |
| [20] |
Qun Lin, Ryan Loxton, Kok Lay Teo. The control parameterization method for nonlinear optimal control: A survey. Journal of Industrial & Management Optimization, 2014, 10 (1) : 275-309. doi: 10.3934/jimo.2014.10.275 |
2018 Impact Factor: 1.292
Tools
Metrics
Other articles
by authors
[Back to Top]