Top

Journal of Scientific Computing

Published in:

05-06-2015

A Stabilized Mixed Finite Element Method for Elliptic Optimal Control Problems

Authors: Hongfei Fu, Hongxing Rui, Jian Hou, Haihong Li

Published in: Journal of Scientific Computing | Issue 3/2016

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

search-config

AI-assisted search

Off

Abstract

In this paper, we propose a new mixed finite element method, called stabilized mixed finite element method, for the approximation of optimal control problems constrained by a first-order elliptic system. This method is obtained by adding suitable elementwise least-squares residual terms for the primal state variable y and its flux \(\sigma \). We prove the coercive and continuous properties for the new mixed bilinear formulation at both continuous and discrete levels. Therefore, the finite element function spaces do not require to satisfy the Ladyzhenkaya–Babuska–Brezzi consistency condition. Furthermore, the state and flux state variables can be approximated by the standard Lagrange finite element. We derive optimality conditions for such optimal control problems under the concept of Discretization-then-Optimization, and then a priori error estimates in a weighted norm are discussed. Finally, numerical experiments are given to confirm the efficiency and reliability of the stabilized method.

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

next article Erratum to: A Stabilized Mixed Finite Element Method for Elliptic Optimal Control Problems

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

Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Society for Industrial and Applied Mathematics, Philadelphia, PA (2002)CrossRef

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

Liu, W., Yan, N.: Adaptive Finite Element Methods for Optimal Control Governed by PDEs, Series in Information and Computational Science, vol. 41. Science Press, Beijing (2008)

Pironneau, O.: Optimal Shape Design for Elliptic Systems. Springer, Berlin (1984)CrossRefMATH

Tiba, D.: Lectures on the Optimal Control of Elliptic Equations. University of Jyvaskyla Press, Finland (1995)

Chen, Y., Liu, W.: Error estimates and superconvergence of mixed finite elements for quadratic optimal control. Int. J. Numer. Anal. Model. 3, 311–321 (2006)MathSciNetMATH

Yan, N., Zhou, Z.: A RT mixed FEM/DG scheme for optimal control governed by convection diffusion equations. J. Sci. Comput. 41, 273–299 (2009)CrossRefMathSciNetMATH

Chen, Y., Lu, Z.: Error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problem. Comput. Methods Appl. Mech. Eng. 199, 1415–1423 (2010)CrossRefMATH

Zhou, J., Chen, Y., Dai, Y.: Superconvergence of triangular mixed finite elements for optimal control problems with an integral constraint. Appl. Math. Comput. 217, 2057–2066 (2010)CrossRefMathSciNetMATH

10.

Fu, H., Rui, H.: A characteristic-mixed finite element method for time-dependent convection–diffusion optimal control problem. Appl. Math. Comput. 218, 3430–3440 (2011)CrossRefMathSciNetMATH

11.

Gong, W., Yan, N.: Mixed finite element methid for Dirichlet boundary control problems governed by elliptic PDEs. SIAM J. Control Optim. 49, 984–1014 (2011)CrossRefMathSciNetMATH

12.

Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. Lecture Notes in Mathematics, vol. 606, pp. 292–315. Springer, Berlin (1977)

13.

Pehlivanov, A.I., Carey, G.F., Lazarov, D.: Least-squares mixed finite elements for second-order elliptic problems. SIAM J. Numer. Anal. 31, 1368–1377 (1994)CrossRefMathSciNetMATH

14.

Cai, Z., Lazarov, R., Manteuffel, T.A., Mccormick, S.F.: First-order system least squares for second-order partial differential equations: part I. SIAM J. Numer. Anal. 31, 1785–1799 (1994)CrossRefMathSciNetMATH

15.

Pani, A.K.: An \(H^1\)-Galerkin mixed finite element method for parabolic partial differential equations. SIAM J. Numer. Anal. 35, 712–727 (1998)CrossRefMathSciNetMATH

16.

Yang, D.P.: A splitting positive defnite mixed element method for miscible displacement ofcompressible flow in porous media. Numer. Methods Part. Differ. Eqn. 17, 229–249 (2001)CrossRefMATH

17.

Rui, H., Kim, S., Kim, S.D.: A remark on least-squares mixed element methods for reaction–diffusion problems. J. Comput. Appl. Math. 202, 230–236 (2007)CrossRefMathSciNetMATH

18.

Gunzburger, M., Lee, H.-C.: A penalty/least-squares method for optimal control problems for first-order elliptic systems. Appl. Math. Comput. 107, 57–75 (2000)CrossRefMathSciNetMATH

19.

Lee, H.-C., Choi, Y.: A least-squares method for optimal control problems for a second-order elliptic system in two dimensions. J. Math. Anal. Appl. 242, 105–128 (2000)CrossRefMathSciNetMATH

20.

Fu, H., Rui, H.: A priori error estimates for least-squares mixed finite element approximation of elliptic optimal control problems. J. Comput. Math. 33, 113–127 (2015)CrossRefMathSciNet

21.

Guo, H., Fu, H., Zhang, J.S.: A splitting positive definite mixed finite element method for elliptic optimal control problems. Appl. Math. Comput. 219, 11178–11190 (2013)CrossRefMathSciNetMATH

22.

Hughes, T., Franca, L.P., Hulbert, G.: A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective–diffusive equations. Comput. Methods Appl. Mech. Eng. 73, 173–189 (1989)CrossRefMathSciNetMATH

23.

Franca, L.P., Stenberg, R.: Error analysis of Galerkin least squares methods for the elasticity equations. SIAM J. Numer. Anal. 28, 1680–1697 (1991)CrossRefMathSciNetMATH

24.

Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)CrossRefMATH

25.

Niceno, B.: http://www-dinma.univ.trieste.it/nirftc/research/easymesh/easymesh.html

26.

Li, R., Liu, W.B.: http://dsec.pku.edu.cn/dsectest

Title: A Stabilized Mixed Finite Element Method for Elliptic Optimal Control Problems
Authors: Hongfei Fu
Hongxing Rui
Jian Hou
Haihong Li
Publication date: 05-06-2015
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 3/2016
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-015-0050-3

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 3/2016

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

Matrix-Free Polynomial-Based Nonlinear Least Squares Optimized Preconditioning and Its Application to Discontinuous Galerkin Discretizations of the Euler Equations

Full 3D Simulations of Two-Phase Core–Annular Flow in Horizontal Pipe Using Level Set Method

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

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

Discontinuous Galerkin Approximation of Linear Parabolic Problems with Dynamic Boundary Conditions

Premium Partner