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Original Articles

A posteriori error estimates for discontinuous Galerkin approximation of non-stationary convection-diffusion optimal control problems

Pages 2106-2123 | Received 10 Apr 2015, Accepted 11 Aug 2015, Published online: 12 Oct 2015

References

  • R. Becker and B. Vexler, Optimal control of the convection-diffusion equation using stabilized finite element methods, Numer. Math. 106 (2007), pp. 349–367. doi: 10.1007/s00211-007-0067-0
  • E. Burman, Crank-Nicolson finite element methods using symmetric stabilization with an application to optimal control problems subject to transient advection-diffusion equations, Commun. Math. Sci. 9(1) (2011), pp. 319–329. doi: 10.4310/CMS.2011.v9.n1.a16
  • A. Cangiani, E.H. Georgoulis, and S. Metcalfe, Adaptive discontinuous Galerkin methods for nonstationary convection-diffusion problems, IMA J. Numer. Anal. 34 (2014), pp. 1578–1597. doi: 10.1093/imanum/drt052
  • L. Chen, iFEM: an innovative finite element methods package in MATLAB. Technical report, Department of Mathematics, University of California, Irvine, Irvine, CA 92697-3875, 2008.
  • A. Demlow, O. Lakkis, and C. Makridakis, A posteriori error estimates in the maximum norm for parabolic problems, SIAM J. Numer. Anal. 47 (2009), pp. 2157–2176. doi: 10.1137/070708792
  • Y. Epshteyn and B. Rivière, Estimation of penalty parameters for symmetric interior penalty Galerkin methods, J. Comput. Appl. Math. 206 (2007), pp. 843–872. doi: 10.1016/j.cam.2006.08.029
  • H.F. Fu and H.X. Rui, A priori error estimates for optimal control problems governed by transient advection-diffusion equations, J. Sci. Comput. 38(3) (2009), pp. 290–315. doi: 10.1007/s10915-008-9224-6
  • H.F. Fu and H.X. Rui, Adaptive characteristic finite element approximation of convection-diffusion optimal control problems, Numer. Meth. PDEs 29 (2013), pp. 979–998. doi: 10.1002/num.21741
  • M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case, Comput. Opti. Appl. 30(1) (2005), pp. 45–61. doi: 10.1007/s10589-005-4559-5
  • T. Katsaounis and I. Kyza, A posteriori error control and adaptivity for Crank-Nicolson finite element approximations for the linear Schrodinger equation, Numer. Math. 129(1) (2015), pp. 55–90. doi: 10.1007/s00211-014-0634-0
  • N. Kopteva and T. Linss, Maximum norm a posteriori error estimation for parabolic problems using elliptic reconstructions, SIAM J. Numer. Anal. 51(3) (2013), pp. 1494–1524. doi: 10.1137/110830563
  • O. Lakkis and Ch. Makridakis, Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems, Math. Comp. 75(256) (2006), pp. 1627–1658. doi: 10.1090/S0025-5718-06-01858-8
  • D. Leykekhman, Investigation of commutative properties of discontinuous Galerkin methods in PDE constrained optimal control problems, J. Sci. Comput. 53(3) (2012), pp. 483–511. doi: 10.1007/s10915-012-9582-y
  • D. Leykekhman and M. Heinkenschloss, Local error analysis of discontinuous Galerkin methods for advection-dominated elliptic linear-quadratic optimal control problems, CAAM Technical Report TR10-11.
  • J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.
  • Ch. Makridakis and R.H. Nochetto, Elliptic reconstruction and a posteriori error estimates for parabolic problems, SIAM J. Numer. Anal. 41 (2003), pp. 1585–1594. doi: 10.1137/S0036142902406314
  • A. Martínez, C. Rodríguez, and M.E. Vázquez-Méndez, Theoretical and numerical analysis of an optimal control problem related to wastewater treatment, SIAM J. Control Optim. 38 (2000), pp. 1534–1553. doi: 10.1137/S0363012998345640
  • G. Murali Mohan Reddy and R.K. Sinha, Ritz-Volterra reconstructions and a posteriori error analysis of finite element method for parabolic integro-differential equations, IMA J. Numer. Anal. 35(1) (2015), pp. 341–371. doi: 10.1093/imanum/drt059
  • D. Schoetzau and L. Zhu, A robust a-posteriori error estimator for discontinuous Galerkin methods for convection-diffusion equations, Appl. Numer. Math. 59(9) (2009), pp. 2236–2255. doi: 10.1016/j.apnum.2008.12.014
  • S. Scott Collis and M. Heinkenschloss, Analysis of the Streamline Upwind/Petrov Galerkin Method applied to the solution of optimal control problems, CAAM TR02-01, Rice University, Houston, TX, 2002.
  • T.J. Sun, Discontinuous Galerkin finite element method with interior penalties for convection diffusion optimal control problem, Int. J. Numer. Anal. Model. 7 (2010), pp. 87–107.
  • N.N. Yan and Z.J. Zhou, A priori and a posteriori error estimates of streamline diffusion finite element method for optimal control problem governed by convection dominated diffusion equation, Numer. Math. Theor. Methods Appl. 1 (2008), pp. 297–320.
  • N.N. Yan and Z.J. Zhou, A priori and a posteriori error analysis of edge stabilization Galerkin method for the optimal control problem governed by convection-dominated diffusion equation, J. Comput. Appl. Math. 223(1) (2009), pp. 198–217. doi: 10.1016/j.cam.2008.01.006
  • H. Yucel, M. Heinkenschloss, and B. Karasozen, Adaptive Symmetric Interior Penalty Galerkin (SIPG) method for optimal control of convection diffusion equations with control constraints, Optimization 63 (2014), pp. 145–166. doi: 10.1080/02331934.2013.801474
  • Z.J. Zhou and N.N. Yan, The local discontinuous Galerkin method for optimal control problem governed by convection diffusion equations, Int. J. Numer. Anal. Model. 7 (2010), pp. 681–699.
  • J. Zhu and Q. Zeng, A mathematical formulation for optimal control of air pollution, Sci. China (D) 46(10) (2003), pp. 994–1002. doi: 10.1007/BF02959394

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