133
Views
20
CrossRef citations to date
0
Altmetric
Section B

Spectral mixed Galerkin method for state constrained optimal control problem governed by the first bi-harmonic equation

&
Pages 2988-3011 | Received 25 Jul 2010, Accepted 11 Feb 2011, Published online: 09 Jun 2011

References

  • Babuška , I. , Osborn , J. and Pitkäranta , J. 1980 . Analysis of Mixed Methods using Mesh-dependent Norms . Math. Comp. , 35 : 1039 – 1079 .
  • Bernardi , C. and Maday , Y. 1997 . “ Spectral methods ” . In Handbook of Numerical Analysis , Edited by: Ciarlet , P. G. and Lions , J.-L. 209 – 486 . Amsterdam : Elsevier .
  • Bernardi , C. , Coppoletta , G. and Maday , Y. 1992 . Some spectral approximations of two-dimensional fourth-order problems . Math. Comput. , 59 : 63 – 76 .
  • Bium , H. and Rannacher , R. 1990 . On mixed finite element methods in plate beding analysis . Comput. Mech. , 6 : 221 – 236 .
  • Brezzi , F. and Raviart , P. A. 1976 . “ Mixed finite element methods for 4-th order elliptic equations ” . In Topics in Numerical Analysis III , Edited by: Miller , J. J.H. 33 – 36 . London : Academic Press .
  • Canuto , C. , Hussaini , M. Y. , Quarteroni , A. and Zang , T. A. 1987 . Spectral Methods in Fluid Dynamics , Berlin : Springer .
  • Cao , W. D. and Yang , D. P. 2009 . Ciarlet-Raviart mixed finite element approximation for an optimal control problem governed by the first bi-harmonic equation . J. Comput. Appl. Math. , 233 ( 2 ) : 372 – 388 .
  • Casas , E. 1986 . Control of an elliptic problem with pointwise state constraints . SIAM J. Control Optim. , 24 ( 4 ) : 1309 – 1322 .
  • Casas , E. and Mateos , M. 2002 . Uniform convergence of the FEM. Applications to state constrained control problems . Comput. Appl. Math. , 21 : 67 – 100 .
  • Casas , E. , Mateos , M. and Tröltzsch , F. 2003 . Necessary and sufficients optimality conditions for optimization problems in function spaces and applications to control theory . ESAIM: Proc. , 13 : 18 – 30 .
  • Charbonneau , A. , Dossou , K. and Pierre , R. 1997 . A residual-based a posteriori error estimator for the Ciarlet–Raviart formulation of the first biharmonic problem . Numer. Meth. Partial Differential Equations , 13 : 93 – 111 .
  • Chen , Z. X. 1997 . Analysis of expand mixed methods for four-order elliptic problems, Numer . Methods Partial Differential Equations , 13 : 483 – 503 .
  • Chen , Y. P. and Liu , W. B. 2006 . Error estimates and superconvergence of mixed finite element for quadratic optimal control . Int. J. Numer. Anal. Model. , 3 ( 3 ) : 311 – 321 .
  • Chen , Y. P. and Liu , W. B. 2008 . A posteriori error estimates for mixed finite element solutions of convex optimal control problems . J. Comput. Appl. Math. , 211 ( 1 ) : 76 – 89 .
  • Chen , Y. , Yi , N. and Liu , W. B. 2008 . A legendre galerkin spectral method for optimal control problems governed by elliptic equations . SIAM J. Numer. Anal. , 46 : 2254 – 2275 .
  • Cheng , X. L. , Han , W. M. and Huang , H. C. 2000 . Some mixed finite element methods for biharmonic equation . J. Comput. Appl. Math. , 126 : 91 – 109 .
  • Hinze , M. and Kunisch , K. 2000 . Optimal and instantaneous control of the instaionary Navier-Stokes equations , Technische Universitat Berlin . Habilitation thesis
  • Lions , J. L. 1971 . Optimal Control of Systems Governed by Partial Differential Equations , Berlin : Springer .
  • Mercier , B. 1974 . Numerical solution of the biharmonic problem by mixed finite element of class C . Boll. Un. Mat. Ital. , 10 ( 1 ) : 133 – 149 .
  • Monk , P. 1987 . A mixed finite element method for the biharmonic equation . SIAM J. Numer. Anal. , 24 ( 4 ) : 737 – 749 .
  • Scholz , R. 1978 . A Mixed method for 4th order problems using linear finite elements . RARIO Anal. Numer. , 33 : 681 – 697 .
  • Shen , J. 1994 . Efficient spectral-galerkin method I. Direct solvers for second and fourth order equations using legendre polynomials . SIAM J. Sci. Comput. , 15 : 1489 – 1505 .
  • Shen , J. 1995 . Efficient spectral-galerkin method II. Direct solvers of second and fourth order equations by using Chebyshev polynomials . SIAM J. Sci. Comput. , 16 : 74 – 87 .
  • Ulibrich , M. 2002 . “ Nonsmooth Newton-like methods for variational inequalities and constrained optimization problems in the function spaces ” . Technische Universitat Munchen . Habilitation thesis
  • Xing , X. Q. and Chen , Y. P. 2008 . Error estimates of mixed methods for optimal control problems governed by parabolic equations . Int. J. Numer Meth. Eng. , 75 : 735 – 754 .
  • Yosida , K. 1978 . Functional Analysis , 5 , Berlin : Springer .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.