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Tianliang Hou1,2, Yanping Chen3
Tianliang Hou, Yanping Chen. MIXED DISCONTINUOUS GALERKIN TIME-STEPPING METHOD FOR LINEAR PARABOLIC OPTIMAL CONTROL PROBLEMS[J]. Journal of Computational Mathematics, 2015, 33(2): 158-178.
[1] N. Arada, E. Casas and F. Tröltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem, Comp. Optim. Appl., 23 (2002), 201-229.[2] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer-Verlag., 95 (1991),65-187.[3] Y. Chen, Superconvergence of quadratic optimal control problems by triangular mixed finite elements, Inter. J. Numer. Meths. Eng., 75 (2008), 881-898.[4] Y. Chen, Y. 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 (2009), 382-403.[5] Y. Chen and T. Hou, Superconvergence and L∞-error estimates of RT1 mixed methods for semilinear elliptic control problems with an integral constraint, Numer. Math. Theor. Meth. Appl. 5 (2012), 423-446.[6] Y. Chen and T. Hou, Error estimates and superconvergence of RT0 mixed methods for a class of semilinear elliptic optimal control problems, Numer. Math. Theor. Meth. Appl., 6 (2013), 637-656.[7] Y. Chen and W. B. Liu, A posteriori error estimates for mixed finite element solutions of convex optimal control problems, J. Comp. Appl. Math., 211 (2008), 76-89.[8] Y. Chen, L. Liu and Z. Lu, A posteriori error estimates of mixed methods for parabolic optimal control problems, Numer. Funct. Anal. Optim., 31 (2010), 1135-1157.[9] J. Douglas and J.E. Roberts, Global estimates for mixed finite element methods for second order elliptic equations, Math. Comp., 44 (1985), 39-52.[10] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem, SIAM J. Numer. Anal., 28 (1991), 43-77.[11] K. Eriksson, C. Johnson and V. Thomée, Time discretization of parabolic problems by the discontinuous Galerkin method, RAIRO Modél. Math. Anal. Numér., 19 (1985), 611-643.[12] W. Gong and N. Yan, A posteriori error estimate for boundary control problems governed by the parabolic partial differential equations, J. Comput. Math., 27 (2009), 68-88.[13] J. Haslinger and P. Neittaanmäki, Finite Element Approximation for Optimal Shape Design, John Wiley and Sons, Chichester, UK, 1989.[14] L. Hou and J.C. Turner, Analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls, Numer. Math., 71 (1995), 289-315.[15] M. Hinze, A variational discretization concept in control constrained optimization: the linearquadratic case, Comp. Optim. Appl., 30 (2005), 45-63.[16] M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with pde constraints. Mathematical Modelling: Theory and Applications, Springer, 2008.[17] P. Houston and E. Süli, A posteriori error analysis for linear convection-diffusion problems under weak mesh regularity assumptions, Report 97/03, Oxford University Computing Laboratory, Oxford, UK, 1997.[18] G. Knowles, Finite element approximation of parabolic time optimal control problems, SIAM J. Control Optim., 20 (1982), 414-427.[19] J. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971.[20] J. Lions and E. Magenes, Non homogeneous boundary value problems and applications, Grandlehre B. 181, Springer-Verlag, 1972.[21] R. Li and W. Liu, http://circus.math.pku.edu.cn/AFEPack.[22] W. Liu, H. Ma, T. Tang, and N. Yan, A posteriori error estimates for discontinuous Galerkin time-stepping method for optimal control problems governed by parabolic equations, SIAM J. Numer. Anal., 42 (2004), 1032-1061.[23] W. Liu and N. Yan, A posteriori error estimates for convex boundary control problems, SIAMJ. Numer. Anal., 39 (2001), 73-99.[24] W. Liu and N. Yan, A posteriori error estimates for optimal control problems governed by Stokes equations, SIAM J. Numer. Anal., 40 (2003), 1850-1869.[25] R. Li, W. Liu, H. Ma, and T. Tang, Adaptive finite element approximation of elliptic control problems, SIAM J. Control Optim., 41 (2002), 1321-1349.[26] C. Meyer and A. Rösch, Superconvergence properties of optimal control problems. SIAM J. Control Optim., 43:3 (2004), 970-985.[27] D. Meidner and B. Vexler, 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 (2008), 1150-1177.[28] D. Meidner and B. Vexler, A priori error estimates for space-time finite element discretization of parabolic optimal control problems Part I: Problems with control constraints, SIAM J. Control Optim., 47 (2008), 1301-1329.[29] R. Mcknight andW. Bosarge, The Ritz-Galerkin procedure for parabolic control problems, SIAM J. Control Optim., 11 (1973), 510-524.[30] P. Neittaanmäki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications, M. Dekker, New York, 1994.[31] P.A. Raviart and J.M. Thomas, A mixed finite element method for 2nd order elliptic problems, Aspecs of the Finite Element Method, Lecture Notes in Math, Springer, Berlin., 606 (1977), 292-315.[32] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, 2nd ed, Springer Ser. Comput. Math. 25, Springer-Verlag, Berlin, 1972.[33] X. Xing and Y. Chen, Error estimates of mixed methods for optimal control problems governed by parabolic equations, Internat. J. Numer. Methods in Engineering., 75 (2008), 735-754. |
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