85
Views
1
CrossRef citations to date
0
Altmetric
Original Articles

Global superconvergence analysis of a nonconforming FEM for Neumann boundary OCPs with elliptic equations

, &
Pages 2451-2461 | Received 25 Mar 2019, Accepted 03 Dec 2019, Published online: 02 Jan 2020

References

  • T. Apel, J. Pfefferer, and A. Rösch, Finite element error estimates for Neumann boundary control problems on graded meshes, Comput. Optim. Appl. 52(1) (2012), pp. 3–28.
  • T. Apel, M. Mateos, J. Pfefferer, and A. Rösch, On the regularity of the solutions of Dirichlet optimal control problems in polygonal domains, SIAM J. Control Optim. 53(6) (2015), pp. 3620–3641.
  • R. Becker, H. Kapp, and R. Rannacher, Adaptive finite element methods for optimal control of partial differential equations: Basic concept, SIAM J. Control Optim. 39(1) (2000), pp. 113–132.
  • J. Bonnans and H. Zidani, Optimal control problems with partially polyhedric constraints, SIAM J. Control Optim. 37(6) (1999), pp. 1726–1741.
  • S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, Berlin, 1994.
  • E. Casas, Error estimates for the numerical approximation of semilinear elliptic control problems with finitely many state constraints, ESAIM: Control Optim. Calc. Var. 8 (2002), pp. 345–374.
  • E. Casas and M. Mateos, Error estimates for the numerical approximation of Neumann control problems, Comput. Optim. Appl. 39(3) (2008), pp. 265–295.
  • Y.P. Chen, Superconvergence of mixed finite element methods for optimal control problems, Math. Comput. 77(263) (2008), pp. 1269–1291.
  • Y.P. Chen, Y.Q. Huang, W.B. Liu, and N.N. Yan, Error estimates and superconvergence of mixed finite element methods for convex optimal control problems, J. Sci. Comput. 42(3) (2011), pp. 382–403.
  • S. Chowdhury, T. Gudi, and A.K. Nandakumaran, A framework for the error analysis of discontinuous finite element methods for elliptic optimal control problems and applications to C0 IP methods, Numer. Funct. Anal. Optim. 36(11) (2015), pp. 1388–1419.
  • J. Douglas Jr., J.E. Santos, D. Sheen, and X. Ye, Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems, ESAIM: Math. Model. Numer. Anal. 33(4) (1999), pp. 747–770.
  • L.C. Evans, Partial Differential Equations, American Mathematical Society, Springer-Verlag, New York, 1998.
  • W. Gong and N.N. Yan, Mixed finite element method for Dirichlet boundary control problem governed by elliptic PDEs, SIAM J. Control Optim. 49(3) (2011), pp. 984–1014.
  • W. Gong and N.N. Yan, Adaptive finite element method for elliptic optimal control problems: Convergence and optimality, Numer. Math. 135(4) (2017), pp. 1121–1170.
  • W. Gong, W.B Liu, Z.Y. Tanet al., A convergent adaptive finite element method for elliptic Dirichlet boundary control problems. IMA J. Numer. Anal. 39(4) (2019), pp. 1985–2015.
  • W. Gong, W.W. Hu, M. Mateos, J. Singler, X. Zhang, and Y.W. Zhang, A new HDG method for Dirichlet boundary control of convection diffusion PDEs II: Low regularity, SIAM. J Numer. Anal. 56(4) (2018), pp. 2262–2287.
  • H.B. Guan and D.Y. Shi, A nonconforming finite element method for constrained optimal control problems governed by parabolic equations, Taiwan. J. Math. 21(5) (2017), pp. 1193–1211.
  • H.B. Guan and D.Y. Shi, An efficient NFEM for optimal control problems governed by a bilinear state equation, Comput. Math. Appl. 77(7) (2019), pp. 1821–1827.
  • H.B. Guan, D.Y. Shi, and X.F. Guan, High accuracy analysis of nonconforming MFEM for constrained optimal control problems governed by Stokes equations, Appl. Math. Lett.53(3) (2016), pp. 17–24.
  • M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case, Comput. Optim. Appl. 30(1) (2005), pp. 45–61.
  • A. Kröner and B. Vexler, A priori error estimates for elliptic optimal control problems with a bilinear state equation, J. Comput. Appl. Math.230 (2009), pp. 781–802.
  • L. Li, A global superconvergent L∞-error estimate of mixed finite element methods for semilinear elliptic optimal control problems, J. Appl. Anal. Comput. 5(3) (2015), pp. 313–328.
  • Q. Lin, L. Tobiska, and A.H. Zhou, Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation, IMA J. Numer. Anal. 25(1) (2005), pp. 160–181.
  • J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.
  • W.B. Liu and N.N. Yan, A posteriori error estimates for convex boundary control problems, SIAM J. Numer. Anal. 39(1) (2001), pp. 73–99.
  • W.B. Liu, W. Gong, and N.N. Yan, A new finite element approximation of a state-constrained optimal control problem, J. Comput. Math. 27(1) (2009), pp. 97–114.
  • M. Mateos and A. Rösch, On saturation effects in the Neumann boundary control of elliptic optimal control problems, Comput. Optim. Appl. 49(2) (2011), pp. 359–378.
  • D.Y. Shi and L.F. Pei, Low order Crouzeix-Raviart type nonconforming finite element methods for approximating Maxwell's equations, Int. J. Numer. Anal. Model. 5(3) (2008), pp. 373–385.
  • D.Y. Shi and C. Xu, EQ1rot nonconforming finite element approximation to Signorini problem, Sci. China Math. 56(6) (2013), pp. 1301–1311.
  • D.Y. Shi, S.P. Mao, and S.C. Chen, An anisotropic nonconforming finite element with some superconvergence results, J. Comput. Math. 23(3) (2005), pp. 261–274.
  • D.Y. Shi, H.H. Wang, and Y.P. Du, An anisotropic nonconforming finite element method for approximating a class of nonlinear Sobolev equations, J. Comput. Math. 27(2–3) (2009), pp. 299–314.
  • D.Y. Shi, C. Xu, and J.H. Chen, Anisotropic nonconforming EQ1rot quadrilateral finite element approximation to second order elliptic problems, J. Sci. Comput. 56(3) (2013), pp. 637–653.
  • D.Y. Shi, J.J. Wang, and F.N. Yan, Unconditional superconvergence analysis for nonlinear parabolic equation with EQ1rot nonconforming finite element, J. Sci. Comput. 70(1) (2017), pp. 85–111.
  • D.Y. Shi, J.J. Wang, and F.N. Yan, Superconvergence analysis for nonlinear parabolic equation with EQ1rot nonconforming finite element, Comput. Appl. Math. 37(1) (2018), pp. 307–327.
  • L. Tobiska and R. Verfürth, Robust a posteriori error estimates for stabilized finite element methods, IMA J. Numer. Anal. 35(4) (2015), pp. 1652–1671.
  • X.J. Xu, On the accuracy of nonconforming quadrilateral Q1 element approximation for the Navier-Stokes problem, SIAM J. Numer. Anal. 38(1) (2000), pp. 17–39.

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.