American Institute of Mathematical Sciences

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March  2018, 8(1): 217-245. doi: 10.3934/mcrf.2018010

Error estimates for Dirichlet control problems in polygonal domains: Quasi-uniform meshes

Thomas Apel 1, , Mariano Mateos 2,, , Johannes Pfefferer 3, and  Arnd Rösch 4,

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

* Corresponding author: Mariano Mateos

Received  April 2017 Revised  September 2017 Published  January 2018

Fund Project: The project was supported by DFG through the International Research Training Group IGDK 1754 Optimization and Numerical Analysis for Partial Differential Equations with Nonsmooth Structures.
The second author was partially supported by the Spanish Ministerio Español de Economía y Competitividad under research projects MTM2014-57531-P and MTM2017-83185-P.

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[11]

M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal., 19 (1988), 613-626.  doi: 10.1137/0519043.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985.  Google Scholar

[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.  Google Scholar

[16]

M. Mateos, Optimization methods for Dirichlet control problems, to appear in Optimization, https://arXiv.org/abs/1701.07619. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J. Nečas, Direct Methods in the Theory of Elliptic Equations, Corrected 2nd edition, Monographs and Studies in Mathematics, Springer Berlin Heidelberg, 2012.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[11]

M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal., 19 (1988), 613-626.  doi: 10.1137/0519043.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985.  Google Scholar

[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.  Google Scholar

[16]

M. Mateos, Optimization methods for Dirichlet control problems, to appear in Optimization, https://arXiv.org/abs/1701.07619. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J. Nečas, Direct Methods in the Theory of Elliptic Equations, Corrected 2nd edition, Monographs and Studies in Mathematics, Springer Berlin Heidelberg, 2012.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

Figure 6.  Constrained problems. Experimental orders of convergence vs biggest angle. Left: generic case. Right: worst case.
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Figure 1.  Convergence rates depending on the maximal interior angle in the unconstrained case
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Figure 2.  Convergence rates depending on the maximal interior angle in the constrained case
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Figure 3.  Family of quasi-uniform meshes which is not $O(h^2)$-irregular
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Figure 4.  Family of quasi-uniform $O(h^2)$-irregular meshes
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Figure 5.  Unconstrained problems. Experimental orders of convergence vs biggest angle.
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