Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Journal of Scientific Computing
Erschienen in: Journal of Scientific Computing 3/2016

04.06.2015

Finite Element Method and A Priori Error Estimates for Dirichlet Boundary Control Problems Governed by Parabolic PDEs

verfasst von: Wei Gong, Michael Hinze, Zhaojie Zhou

Erschienen in: Journal of Scientific Computing | Ausgabe 3/2016

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

Finite element approximations of Dirichlet boundary control problems governed by parabolic PDEs on convex polygonal domains are studied in this paper. The existence of a unique solution to optimal control problems is guaranteed based on very weak solution of the state equation and \(L^2(0,T;L^2(\varGamma ))\) as control space. For the numerical discretization of the state equation we use standard piecewise linear and continuous finite elements for the space discretization of the state, while a dG(0) scheme is used for time discretization. The Dirichlet boundary control is realized through a space–time \(L^2\)-projection. We consider both piecewise linear, continuous finite element approximation and variational discretization for the controls and derive a priori \(L^2\)-error bounds for controls and states. We finally present numerical examples to support our theoretical findings.
Vorheriger Artikel Matrix-Free Polynomial-Based Nonlinear Least Squares Optimized Preconditioning and Its Application to Discontinuous Galerkin Discretizations of the Euler Equations
Nächster Artikel A Stabilized Mixed Finite Element Method for Elliptic Optimal Control Problems

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

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!

Jetzt informieren

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!

Jetzt informieren

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!

Jetzt informieren
Literatur
1.
Apel, T., Flaig, T.G.: Crank–Nicolson schemes for optimal control problems with evolution equations. SIAM J. Numer. Anal. 50, 1484–1512 (2012)CrossRefMathSciNetMATH
2.
Arada, N., Raymond, J.P.: Dirichlet boundary control of semilinear parabolic equations: part I. Appl. Math. Optim. 45, 125–143 (2002)CrossRefMathSciNetMATH
3.
Belgacem, F.B., Bernardi, C., Fekih, H.E.: Dirichlet boundary control for a parabolic equation with a final observation I: a space–time mixed formulation and penalization. Asymptot Anal. 71, 101–121 (2011)
4.
5.
Bramble, J.H., Pasciak, J.E., Schatz, A.H.: The construction of preconditioners for elliptic problems by substructing I. Math. Comput. 47, 103–134 (1986)CrossRefMathSciNetMATH
6.
Casas, E., Mateos, M., Raymond, J.P.: Penalization of Dirichlet optimal control problems. ESAIM Control Optim. Calc. Var. 15, 782–809 (2009)CrossRefMathSciNetMATH
7.
Casas, E., Raymond, J.P.: Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations. SIAM J. Control Optim. 45, 1586–1611 (2006)CrossRefMathSciNetMATH
8.
Casas, E., Sokolowski, J.: Approximation of boundary control problems on curved domains. SIAM J. Control Optim. 48, 3746–3780 (2010)CrossRefMathSciNetMATH
9.
Ciarlet, P.G.: The Finite Element Methods for Elliptic Problems. Elsevier, North-Holland (1978)
10.
Deckelnick, K., Günther, A., Hinze, M.: Finite element approximation of Dirichlet boundary control for elliptic PDEs on two- and three-dimensional curved domains. SIAM J. Control Optim. 48, 2798–2819 (2009)CrossRefMathSciNetMATH
11.
French, D.A., King, J.T.: Approximation of an elliptic control problem by the finite element method. Numer. Funct. Anal. Optim. 12, 299–314 (1991)CrossRefMathSciNetMATH
12.
French, D.A., King, J.T.: Analysis of a robust finite element approximation for a parabolic equation with rough boundary data. Math. Comput. 60, 79–104 (1993)CrossRefMathSciNetMATH
13.
Fursikov, A.V., Gunzburger, M.D., Hou, L.S.: Boundary value problems and optimal boundary control for the Navier-Stokes systems: the two-dimensional case. SIAM J. Control Optim. 36, 852–894 (1998)CrossRefMathSciNetMATH
14.
Geveci, T.: On the approximation of the solution of an optimal control problem governed by an elliptic equation. RAIRO Anal. Numer. 13, 313–328 (1979)MathSciNetMATH
15.
Gong, W., Yan, N.N.: A posteriori error estimate for boundary control problems governed by the parabolic partial differential equations. J. Comput. Math. 27, 68–88 (2009)MathSciNetMATH
16.
Gong, W., Yan, N.N.: Mixed finite element method for Dirichlet boundary control problem governed by elliptic PDEs. SIAM J. Control Optim. 49, 984–1014 (2011)CrossRefMathSciNetMATH
17.
Grisvard, P.: Singularities in Boundary Value Problems. Springer, Berlin (1992)MATH
18.
Gunzburger, M.D., Hou, L.S.: Treating inhomogeneous essential boundary conditions in finite element methods and the calculation of boundary stresses. SIAM J. Numer. Anal. 29, 390–424 (1992)CrossRefMathSciNetMATH
19.
Hinze, M.: A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 30, 45–63 (2005)CrossRefMathSciNetMATH
20.
Hinze, M., Kunisch, K.: Second order methods for boundary control of the instationary Navier–Stokes system. ZAMM Z. Angew. Math. Mech. 84, 171–187 (2004)CrossRefMathSciNetMATH
21.
Hinze, M., Matthes, U.: A note on variational discretization of Neumann boundary control problems. Control Cybern. 38, 577–591 (2009)MathSciNetMATH
22.
Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints, Mathematical Modelling: Theory and Applications, vol. 23. Springer, Berlin (2009)
23.
Kunisch, K., Vexler, B.: Constrained Dirichlet boundary control in \(L^2\) for a class of evolution equations. SIAM J. Control Optim. 46, 1726–1753 (2007)CrossRefMathSciNetMATH
24.
Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)CrossRefMATH
25.
Lions, J.L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications, vol. I, II. Springer, Berlin (1972)CrossRef
26.
Liu, W.B., Ma, H.P., Tang, T., Yan, N.N.: A posteriori error estimates for discontinuous Galerkin time-stepping method for optimal control problems governed by parabolic equations. SIAM J. Numer. Anal. 42, 1032–1061 (2004)CrossRefMathSciNetMATH
27.
Liu, W.B., Yan, N.N.: A posteriori error estimates for optimal control problems governed by parabolic equations. Numer. Math. 93, 497–521 (2003)CrossRefMathSciNetMATH
28.
Liu, W.B., Yan, N.N.: Adaptive Finite Element Methods for Optimal Control Governed by PDEs. Science press, Beijing (2008)
29.
May, S., Rannacher, R., Vexler, B.: Error analysis fo a finite element approximation of elliptic Dirichlet boundary control problems. SIAM J. Control Optim. 51(3), 2585–2611 (2013)CrossRefMathSciNetMATH
30.
Meidner, D., Vexler, B.: A priori error estimates for space–time finite element discretization of parabolic optimal control problems. Part I: problems without control constraints. SIAM J. Control Optim. 47(3), 1150–1177 (2008)CrossRefMathSciNetMATH
31.
Meidner, D., Vexler, B.: A priori error estimates for space–time finite element discretization of parabolic optimal control problems. Part II: problems with control constraints. SIAM J. Control Optim. 47(3), 1301–1329 (2008)CrossRefMathSciNetMATH
32.
Meidner, D., Vexler, B.: A priori error analysis of the Petrov-Galerkin Crank–Nicolson scheme for parabolic optimal control problems. SIAM J. Control Optim. 49(5), 2183–2211 (2011)CrossRefMathSciNetMATH
33.
Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (2006)MATH
34.
Vexler, B.: Finite element approximation of elliptic Dirichlet optimal control problems. Numer. Funct. Anal. Optim. 28, 957–973 (2007)CrossRefMathSciNetMATH
35.
von Daniels, N., Hinze, M., Vierling, M.: Crank-Nicolson time stepping and variational discretization of control-constrained parabolic optimal control problems. SIAM J. Control Optim. 53(3), 1182–1198 (2015)
36.
Winther, R.: Error estimates for a Galerkin approximation of a parabolic control problem. Ann. Math. Pura App. 117(4), 173–206 (1978)CrossRefMathSciNetMATH
Metadaten
Titel
Finite Element Method and A Priori Error Estimates for Dirichlet Boundary Control Problems Governed by Parabolic PDEs
verfasst von
Wei Gong
Michael Hinze
Zhaojie Zhou
Publikationsdatum
04.06.2015
Verlag
Springer US
Erschienen in
Journal of Scientific Computing / Ausgabe 3/2016
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-015-0051-2

Weitere Artikel der Ausgabe 3/2016

Journal of Scientific Computing 3/2016 Zur Ausgabe

OriginalPaper

Direct Minimization for Ensemble Electronic Structure Calculations

OriginalPaper

Second Order Conformal Symplectic Schemes for Damped Hamiltonian Systems

OriginalPaper

Discontinuous Galerkin Approximation of Linear Parabolic Problems with Dynamic Boundary Conditions

OriginalPaper

Well-Balanced High Order 1D Schemes on Non-uniform Grids and Entropy Residuals

OriginalPaper

Two Alternating Direction Implicit Difference Schemes for Two-Dimensional Distributed-Order Fractional Diffusion Equations

OriginalPaper

Numerical Algorithms with High Spatial Accuracy for the Fourth-Order Fractional Sub-Diffusion Equations with the First Dirichlet Boundary Conditions