Top

Journal of Scientific Computing

Published in:

06-04-2017

Convergence and Quasi-Optimality of an Adaptive Finite Element Method for Optimal Control Problems on \(L^{2}\) Errors

Authors: Haitao Leng, Yanping Chen

Published in: Journal of Scientific Computing | Issue 1/2017

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

In this paper, we prove the convergence of an adaptive finite element method for optimal control problems on \(L^{2}\) errors by keeping the meshes sufficiently mildly. In order to keep the meshes sufficiently mildly we need increasing the number of elements that are refined, moreover, we find that it will not compromise the quasi-optimality of the AFEM. In other words, we prove the quasi-optimality of the adaptive finite element algorithm in the present paper. In the end, we conclude this paper with some conclusions and future works.

previous article An Efficient Primal-Dual Method for the Obstacle Problem

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Babus̃ka, I., Rheinboldt, W.C.: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15, 736–754 (1978)MathSciNetCrossRefMATH

Dörfler, W.: A convergent adaptive algorithm for Poissons equation. SIAM J. Numer. Anal. 33, 1106–1124 (1996)MathSciNetCrossRefMATH

Morin, P., Nochetto, R.H., Siebert, K.G.: Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38, 466–488 (2000)MathSciNetCrossRefMATH

Morin, P., Nochetto, R.H., Siebert, K.G.: Convergence of adaptive finite element methods. SIAM Rev. 44, 631–658 (2002)MathSciNetCrossRefMATH

Morin, P., Nochetto, R.H., Siebert, K.G.: Local problems on stars: a posteriori error estimates convergence, and performance. Math. Comput. 72, 1067–1097 (2003)CrossRefMATH

Morin, P., Siebert, K.G., Veeser, A.: A basic convergence result for conforming adaptive finite elements. Math. Models Methods Appl. 18, 707–737 (2008)MathSciNetCrossRefMATH

Chen, Z., Feng, J.: An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems. Math. Comput. 73, 1167–1193 (2004)MathSciNetCrossRefMATH

Mekchay, K., Nochetto, R.H.: Convergence of adaptive finite element methods for general second order linear elliptic PDEs. SIAM J. Numer. Anal. 43, 1803–1827 (2005)MathSciNetCrossRefMATH

Cascon, J.M., Kreuzer, C., Nochetto, R.H., Siebert, K.G.: Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46, 2524–2550 (2008)MathSciNetCrossRefMATH

10.

Binev, P., Dahmen, W., Devore, R.: Adaptive finite element methods with convergence rates. Numer. Math. 9, 219–268 (2004)MathSciNetCrossRefMATH

11.

Stevenson, R.: Optimality of a standard adaptive finite element method. Found. Comput. Math. 7, 245–269 (2007)MathSciNetCrossRefMATH

12.

Chen, L., Holst, M., Xu, J.C.: Convergence and optimality of adaptive mixed finite element methods. Math. Comput. 78, 35–53 (2009)MathSciNetCrossRefMATH

13.

Carstensen, C., Hoppe, R.H.W.: Error reduction and convergence for an adaptive mixed finite element method. Math. Comput. 75, 1033–1042 (2006)MathSciNetCrossRefMATH

14.

Carstensen, C., Gallistl, D., Schedensack, M.: Quasi-optimal adaptive pseudostress approxoimation of the Stokes equations. SIAM J. Numer. Anal. 51, 1715–1734 (2013)MathSciNetCrossRefMATH

15.

Becker, R., Mao, S.P., Shi, Z.C.: A convergence nonconforming adaptive finite element method with quasi-optimal complexity. SIAM J. Numer. Anal. 47, 4639–4659 (2010)MathSciNetCrossRefMATH

16.

Becker, R., Mao, S.P.: Quasi-optimality of adaptive nonconforming finite element methods for the Stokes equations. SIAM J. Numer. Anal. 49, 970–991 (2011)MathSciNetCrossRefMATH

17.

Becker, R., Kapp, H., Rannacher, R.: Adaptive finite element methods for optimal control of partial differential equations: basic concept. SIAM J. Control Optim. 39, 113–132 (2000)MathSciNetCrossRefMATH

18.

Demlow, A., Stevenson, R.: Convergence and quasi-optimality of an adaptive finite element method for controlling \(L^{2}\) errors. Numer. Math. 117, 185–218 (2011)MathSciNetCrossRefMATH

19.

Liu, W.B., Yan, N.N.: A posteriori error analysis for convex distributed optimal control problems. Adv. Comp. Math. 15, 285–309 (2001)CrossRefMATH

20.

Liu, W.B., Yan, N.N.: A posteriori error estimates for convex boundary control problems. SIAM J. Numer. Anal. 39, 73–99 (2001)MathSciNetCrossRefMATH

21.

Liu, W.B., Yan, N.N.: A posteriori error estimates for optimal problems governed by Stokes equations. SIAM J. Numer. Anal. 40, 1850–1869 (2003)CrossRefMATH

22.

Chen, Y., Liu, W.B.: A posteriori error estimates for mixed finite element solutions of convex optimal control problems. J. Comput. Appl. Math. 211, 76–89 (2008)MathSciNetCrossRefMATH

23.

Chen, Y., Liu, L.L., Lu, Z.L.: A posteriori error estimates of mixed methods for parabolic optimal control problems. Numer. Funct. Anal. Optim. 31, 1135–1157 (2010)MathSciNetCrossRefMATH

24.

Hou, T.L., Chen, Y., Huang, Y.Q.: A posteriori error estimates of mixed methods of quadratic optimal control problems governed by parabolic equations. Numer. Math. Theory Methods Appl. 4, 439–458 (2011)MathSciNetMATH

25.

Li, R., Liu, W.B., Ma, H.P., Tang, T.: Adaptive finite element approximation for distributed elliptic optimal control problems. SIAM J. Control Optim. 41, 1321–1349 (2002)MathSciNetCrossRefMATH

26.

Gaevskaya, A., Hoppe, R.H.W., Iliash, Y., Kieweg, M.: Convergence analysis of an adaptive finite element method for distributed control problems with control constraints. Int. Ser. Numer. Math. 155, 47–68 (2007)MathSciNetMATH

27.

Kohls, K., Rösch, A., Siebert, K.G.: Convergence of adaptive finite elements for control constraints optimal control problems, preprint-Nr.: SPP1253-153 (2013)

28.

Kohls, K., Rösch, A., Siebert, K.G.: A posteriori error analysis of optimal control problems with control constraints. SIAM J. Control Optim. 52, 1832–1861 (2014)MathSciNetCrossRefMATH

29.

Gong, W., Yan, N.N.: Adaptive finite element method for elliptic optimal control problems: convergence and optimality, preprint arXiv: 1505.01632 (2015)

30.

Verfürth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, Chichester (1996)MATH

31.

Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)CrossRefMATH

32.

Ge, L., Liu, W.B., Yang, D.P.: \(L^{2}\) norm equivalent a posteriori error estimate for a constrained optiaml control problem. Int. J. Numer. Anal. Model. 6, 335–353 (2009)MathSciNetMATH

Title: Convergence and Quasi-Optimality of an Adaptive Finite Element Method for Optimal Control Problems on Errors
Authors: Haitao Leng
Yanping Chen
Publication date: 06-04-2017
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 1/2017
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-017-0425-8

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Other articles of this Issue 1/2017

Spectral Method for Vorticity-Stream Function Form of Navier–Stokes Equations in an Infinite Channel with Slip Boundary Conditions

A Block-Centered Finite Difference Method for Slightly Compressible Darcy–Forchheimer Flow in Porous Media

Computation of Time Optimal Control Problems Governed by Linear Ordinary Differential Equations

The Time Second Order Mass Conservative Characteristic FDM for Advection–Diffusion Equations in High Dimensions

Hamiltonian Finite Element Discretization for Nonlinear Free Surface Water Waves

A Finite Element Method for High-Contrast Interface Problems with Error Estimates Independent of Contrast

Premium Partner