Figure 6.
Constrained problems. Experimental orders of convergence vs biggest angle. Left: generic case. Right: worst case.
-
Previous Article
Optimal control of a non-smooth semilinear elliptic equation
- MCRF Home
- This Issue
-
Next Article
Error analysis for global minima of semilinear optimal control problems
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 and 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 and Management Optimization, 2013, 9 (3) : 631-642. doi: 10.3934/jimo.2013.9.631 |
| [2] |
Dominik Hafemeyer, Florian Mannel, Ira Neitzel, Boris Vexler. Finite element error estimates for one-dimensional elliptic optimal control by BV-functions. Mathematical Control and Related Fields, 2020, 10 (2) : 333-363. doi: 10.3934/mcrf.2019041 |
| [3] |
Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control and Related Fields, 2021, 11 (3) : 601-624. doi: 10.3934/mcrf.2021014 |
| [4] |
Shaolin Ji, Xiaole Xue. A stochastic maximum principle for linear quadratic problem with nonconvex control domain. Mathematical Control and Related Fields, 2019, 9 (3) : 495-507. doi: 10.3934/mcrf.2019022 |
| [5] |
Zhen-Zhen Tao, Bing Sun. Error estimates for spectral approximation of flow optimal control problem with $ L^2 $-norm control constraint. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022030 |
| [6] |
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 |
| [7] |
Konstantinos Chrysafinos, Efthimios N. Karatzas. Symmetric error estimates for discontinuous Galerkin approximations for an optimal control problem associated to semilinear parabolic PDE's. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1473-1506. doi: 10.3934/dcdsb.2012.17.1473 |
| [8] |
Quyen Tran, Harbir Antil, Hugo Díaz. Optimal control of parameterized stationary Maxwell's system: Reduced basis, convergence analysis, and a posteriori error estimates. Mathematical Control and Related Fields, 2022 doi: 10.3934/mcrf.2022003 |
| [9] |
Roberto Triggiani, Xiang Wan. From low to high-and lower-optimal regularity of the SMGTJ equation with Dirichlet and Neumann boundary control, and with point control, via explicit representation formulae. Evolution Equations and Control Theory, 2022 doi: 10.3934/eect.2022007 |
| [10] |
Luca Di Persio, Giacomo Ziglio. Gaussian estimates on networks with applications to optimal control. Networks and Heterogeneous Media, 2011, 6 (2) : 279-296. doi: 10.3934/nhm.2011.6.279 |
| [11] |
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 and Related Fields, 2018, 8 (2) : 411-449. doi: 10.3934/mcrf.2018017 |
| [12] |
Simone Göttlich, Patrick Schindler. Optimal inflow control of production systems with finite buffers. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 107-127. doi: 10.3934/dcdsb.2015.20.107 |
| [13] |
Ahmad Ahmad Ali, Klaus Deckelnick, Michael Hinze. Error analysis for global minima of semilinear optimal control problems. Mathematical Control and Related Fields, 2018, 8 (1) : 195-215. doi: 10.3934/mcrf.2018009 |
| [14] |
Alexander Arguchintsev, Vasilisa Poplevko. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control and Optimization, 2018, 8 (2) : 193-202. doi: 10.3934/naco.2018011 |
| [15] |
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 and Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025 |
| [16] |
Ciro D’Apice, Umberto De Maio, Peter I. Kogut. Boundary velocity suboptimal control of incompressible flow in cylindrically perforated domain. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 283-314. doi: 10.3934/dcdsb.2009.11.283 |
| [17] |
Marc Puche, Timo Reis, Felix L. Schwenninger. Funnel control for boundary control systems. Evolution Equations and Control Theory, 2021, 10 (3) : 519-544. doi: 10.3934/eect.2020079 |
| [18] |
Tan H. Cao, Boris S. Mordukhovich. Applications of optimal control of a nonconvex sweeping process to optimization of the planar crowd motion model. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4191-4216. doi: 10.3934/dcdsb.2019078 |
| [19] |
Fabio Bagagiolo. Optimal control of finite horizon type for a multidimensional delayed switching system. Discrete and Continuous Dynamical Systems - B, 2005, 5 (2) : 239-264. doi: 10.3934/dcdsb.2005.5.239 |
| [20] |
Chunjuan Hou, Yanping Chen, Zuliang Lu. Superconvergence property of finite element methods for parabolic optimal control problems. Journal of Industrial and Management Optimization, 2011, 7 (4) : 927-945. doi: 10.3934/jimo.2011.7.927 |
2020 Impact Factor: 1.284
Tools
Metrics
Other articles
by authors
[Back to Top]