Abstract
In this paper, we study the virtual element discretization of an elliptic optimal control problem with pointwise Neumann boundary control constraint. We construct a virtual element discrete scheme using virtual element discretzation of the state equation and variational discretization of the control variable. A priori error estimates for the state, adjoint state and control variable with respect to \(H^1\) and \(L^2\) norm are derived. Finally, numerical examples are presented to verify the theoretical findings.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adams RA, Fournier JJF (2003) Sobolev spaces. Elsevier, Amsterdam, pp 62–65
Ahmad B, Alsaedi A, Brezzi F et al (2013) Equivalent projectors for virtual element methods. Computers Mathematics with Applications 66(3):376–391. https://doi.org/10.1016/j.camwa.2013.05.015
Apel T, Pfefferer J, Rösch A (2012) Finite element error estimates for Neumann boundary control problems on graded meshes. Comput Optim Appl 52(1):3–28. https://doi.org/10.1007/s10589-011-9427-x
Apel T, Pfefferer J, Rösch A (2015) Finite element error estimates on the boundary with application to optimal control. Math Comput 84(291):33–70. https://doi.org/10.1090/S0025-5718-2014-02862-7
Apel T, Winkler M, Pfefferer J (2018) Error estimates for the postprocessing approach applied to Neumann boundary control problems in polyhedral domains. IMA J Numer Anal 38(4):1984–2025. https://doi.org/10.1093/imanum/drx059
Beirão da Veiga L, Brezzi F, MariniL D et al (2016) Virtual element method for general second-order elliptic problems on polygonal meshes. Math Models Methods Appl Sci 26(04):729–750. https://doi.org/10.1142/S0218202516500160
BeirãodaVeiga L, Brezzi F, Cangiani A et al (2013) Basic principles of virtual element methods. Math Models Methods Appl Sci 23(01):199–214. https://doi.org/10.1142/S0218202512500492
Benedetto MF, Berrone S, Borio A et al (2016) A hybrid mortar virtual element method for discrete fracture network simulations. J Comput Phys 306:148–166. https://doi.org/10.1016/j.jcp.2015.11.034
Brenner SC, Scott LR, Scott LR (2008) The mathematical theory of finite element methods. Springer, New York
Brenner SC, Sung LY, Tan Z (2021) A \(C^1\) virtual element method for an elliptic distributed optimal control problem with pointwise state constraints. Math Models Methods Appl Sci 31(14):2887–2906. https://doi.org/10.1142/S0218202521500640
Cangiani A, Georgoulis EH, Pryer T et al (2017) A posteriori error estimates for the virtual element method. Numer Math 137:857–893. https://doi.org/10.1007/s00211-017-0891-9
Cangiani A, Manzini G, Sutton OJ (2017) Conforming and nonconforming virtual element methods for elliptic problems. IMA J Numer Anal 37(3):1317–1354. https://doi.org/10.1093/imanum/drw036
Casas E, Mateos M (2008) Error estimates for the numerical approximation of Neumann control problems. Comput Optim Appl 39(3):265–295. https://doi.org/10.1007/s10589-005-2180-2
Casas E, Mateos M, Tröltzsch F (2005) Error estimates for the numerical approximation of boundary semilinear elliptic control problems. Comput Optim Appl 31(2):193–219. https://doi.org/10.1007/s10589-007-9056-6
da Veiga LB, Manzini G (2014) A virtual element method with arbitrary regularity. IMA J Numer Anal 34(2):759–781. https://doi.org/10.1093/imanum/drt018
Da Veiga LB, Brezzi F, Marini LD (2013) Virtual elements for linear elasticity problems. SIAM J Numer Anal 51(2):794–812. https://doi.org/10.1137/120874746
Frittelli M, Sgura I (2018) Virtual element method for the Laplace-Beltrami equation on surfaces. ESAIM: Mathematical Modelling and Numerical Analysis 52(3):965–993. https://doi.org/10.1051/m2an/2017040
Frittelli M, Madzvamuse A, Sgura I (2021) Bulk-surface virtual element method for systems of PDEs in two-space dimensions. Numer Math 147(2):305–348. https://doi.org/10.1007/s00211-020-01167-3
Geveci T (1979) On the approximation of the solution of an optimal control problem governed by an elliptic equation. RAIRO. Analyse numérique 13(4):313–328
Grenkin GV, Chebotarev AY, Kovtanyuk AE et al (2016) Boundary optimal control problem of complex heat transfer model. J Math Anal Appl 433(2):1243–1260. https://doi.org/10.1016/j.jmaa.2015.08.049
Hinze M, Matthes U (2009) A note on variational discretization of elliptic Neumann boundary control. Control Cybernetics, 38(3): 577-591. http://eudml.org/doc/209630
Kinderlehrer D, Stampacchia G (2000) An introduction to variational inequalities and their applications. Society for Industrial and Applied Mathematics 35–39
Krumbiegel K, Pfefferer J (2015) Superconvergence for Neumann boundary control problems governed by semilinear elliptic equations. Comput Optim Appl 61(2):373–408. https://doi.org/10.1007/s10589-014-9718-0
Lee H (2011) Optimal control for quasi-Newtonian flows with defective boundary conditions. Comput Methods Appl Mech Eng 200(33–36):2498–2506. https://doi.org/10.1016/j.cma.2011.04.019
Leng H, Chen Y (2021) Residual-type a posteriori error analysis of HDG methods for Neumann boundary control problems. Adv Comput Math 47(3):1–20. https://doi.org/10.1007/s10444-021-09864-9
Liu H, Yan N (2006) Superconvergence and a posteriori error estimates for boundary control governed by Stokes equations. Journal of Computational Mathematics, 2006: 343-356. https://www.jstor.org/stable/43693295
Metzger M (2001) Optimal control of crystal growth processes. J Cryst Growth 230(1–2):210–216. https://doi.org/10.1016/S0022-0248(01)01343-4
Park K, Chi H, Paulino GH (2019) On nonconvex meshes for elastodynamics using virtual element methods with explicit time integration. Comput Methods Appl Mech Eng 356:669–684. https://doi.org/10.1016/j.cma.2019.06.031
Pfefferer J (2014) Numerical analysis for elliptic Neumann boundary control problems on polygonal domains. München, Univ. der Bundeswehr, Diss, 2014
Pingaro M, Reccia E, Trovalusci P et al (2019) Fast statistical homogenization procedure (FSHP) for particle random composites using virtual element method. Comput Mech 64(1):197–210. https://doi.org/10.1007/s00466-018-1665-7
Wang Q, Zhou Z (2021) Adaptive virtual element method for optimal control problem governed by general elliptic equation. J Sci Comput 88(1):1–33. https://doi.org/10.1007/s10915-021-01528-6
Wang Q, Zhou Z (2022) A priori and a posteriori error analysis for virtual element discretization of elliptic optimal control problem. Numerical Algorithms 90(3):989–1015. https://doi.org/10.1007/s11075-021-01219-1
Yu H, Liu B (2013) Optimal control of backward stochastic heat equation with Neumann boundary control and noise. Stochastics An International Journal of Probability and Stochastic Processes 85(3):532–558. https://doi.org/10.1080/17442508.2011.654345
Zhu J, Zeng Q, Guo D et al (1997) Optimal control problems related to the navigation channel engineering. Sci China Ser E: Technol Sci 40:82–88. https://doi.org/10.1007/BF02916593
Acknowledgements
The research was supported by the National Natural Science Foundation of China under Grant Nos. 11971276 and 12171287.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.
Additional information
Communicated by Frederic Valentin.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Liu, S., Zhou, Z. Virtual element method for elliptic Neumann boundary optimal control problem. Comp. Appl. Math. 42, 142 (2023). https://doi.org/10.1007/s40314-023-02282-1
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-023-02282-1