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Symmetric error estimates for discontinuous Galerkin approximations for an optimal control problem associated to semilinear parabolic PDE's
1. | National Technical University of Athens, School of Applied Mathematical and Physical Sciences, Department of Mathematics, Zografou Campus, 15780, Athens, Greece, Greece |
References:
[1] |
G. Akrivis and C. Makridakis, Galerkin time-stepping methods for nonlinear parabolic equations,, M2AN Math. Model. and Numer. Anal., 38 (2004), 261.
doi: 10.1051/m2an:2004013. |
[2] |
G. Allaire and O. Pantz, Structural optimization with FreeFem++,, Struct. Multidiscip. Optim., 32 (2006), 173.
doi: 10.1007/s00158-006-0017-y. |
[3] |
T. Apel and T. Flaig, Crank-Nicolson schemes for optimal control problems with evolution equations,, submitted. Available from: \url{http://www.unibw.de/bauv1/personen/apel/papers}., (). Google Scholar |
[4] |
A. Borzi and R. Griesse, Distributed optimal control for lambda-omega systems,, J. Numer. Math., 14 (2006), 17.
doi: 10.1515/156939506776382120. |
[5] |
E. Casas and J.-P. Raymond, Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equation,, SIAM J. Control and Optim., 45 (2006), 1586.
doi: 10.1137/050626600. |
[6] |
E. Casas, M. Mateos and F. Tröltzsch, Error estimates for the numerical approximation of boundary semilinear elliptic control problem,, Comput. Optim. and Appl., 31 (2005), 193.
|
[7] |
K. Chrysafinos, Discontinous Galerkin approximations for distributed optimal control problems constrained by parabolic PDE's,, Int. J. Numer. Anal. and Mod., 4 (2007), 690.
|
[8] |
K. Chrysafinos, Analysis and finite element approximations for distributed optimal control problems for implicit parabolic equations,, J. Comput. Appl. Math., 231 (2009), 327.
doi: 10.1016/j.cam.2009.02.092. |
[9] |
K. Chrysafinos, Convergence of discontinuous Galerkin approximations of an optimal control problem associated to semilinear parabolic PDE's,, M2AN Math. Model. Numer. Anal., 44 (2010), 189.
|
[10] |
K. Chrysafinos, Convergence of discontinuous time-stepping schemes for a Robin boundary control problem under minimal regularity assumptions,, submitted. Available from: \url{http://www.math.ntua.gr/~chrysafinos}., (). Google Scholar |
[11] |
K. Chrysafinos and N. J. Walkington, Error estimates for the discontinuous Galerkin methods for parabolic equations,, SIAM J. Numer. Anal., 44 (2006), 349.
doi: 10.1137/030602289. |
[12] |
K. Chrysafinos and N. J. Walkington, Discontinuous Galerkin approximations of the Stokes and Navier-Stokes equations,, Math. Comp., 79 (2010), 2135.
doi: 10.1090/S0025-5718-10-02348-3. |
[13] |
K. Chrysafinos and N. J. Walkington, Lagrangian and moving mesh methods for the convection diffusion equation,, M2AN Math. Model. Numer. Anal., 42 (2008), 25.
|
[14] |
K. Chrysafinos, M. D. Gunzburger and L. S. Hou, Semidiscrete approximations of optimal Robin boundary control problems constrained by semilinear parabolic PDE,, J. Math. Anal. Appl., 323 (2006), 891.
doi: 10.1016/j.jmaa.2005.10.053. |
[15] |
P. G. Ciarlet, "The Finite Element Method for Elliptic Problems,", Reprint of the 1978 original, (1978).
|
[16] |
K. Deckelnick and M. Hinze, Semidiscretization and error estimates for distributed control of the instationary Navier-Stokes equations,, Numer. Math., 97 (2004), 297.
doi: 10.1007/s00211-003-0507-4. |
[17] |
K. Deckelnick and M. Hinze, Variational discretization of parabolic control problems in the presence of pointwise state constraints,, J. Comput. Math., 29 (2011), 1.
|
[18] |
T. F. Dupont and Y. Liu, Symmetric error estimates for moving mesh Galerkin methods for advection-diffusion equations,, SIAM J. Numer. Anal., 40 (2002), 914.
doi: 10.1137/S0036142900380431. |
[19] |
D. Estep and S. Larsson, The discontinuous Galerkin method for semilinear parabolic problems,, RAIRO Modél. Math. Anal. Numér., 27 (1993), 35.
|
[20] |
K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem,, SIAM J. Numer. Anal., 28 (1991), 43.
doi: 10.1137/0728003. |
[21] |
K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. II. Optimal error estimates in $L_{\infty}(L^2)$ and $L_{\infty}(L_{\infty})$,, SIAM J. Numer. Anal., 32 (1995), 706.
doi: 10.1137/0732033. |
[22] |
K. Ericksson and C. Johnson, Adaptive finite element methods for parabolic problems IV, Nonlinear problems,, SIAM J. Numer. Anal., 32 (1995), 1729.
doi: 10.1137/0732078. |
[23] |
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.
|
[24] |
L. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (1998).
|
[25] |
R. Falk, Approximation of a class of otimal control problems with order of convergence estimates,, J. Math. Anal. Appl., 44 (1973), 28.
doi: 10.1016/0022-247X(73)90022-X. |
[26] |
A. Fursikov, "Optimal Control of Distributed Systems. Theory and Applications,", Translations of Mathematical Monographs, 187 (2000).
|
[27] |
V. Girault and P.-A. Raviart, "Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms,", Springer Series in Computational Mathematics, 5 (1986).
|
[28] |
W. Gong, M. Hinze and Z. Zhou, A priori error analysis for finite element approximation of parabolic optimal control problems with pointwise control,, submitted. Available from: \url{http://preprint.math.uni-hamburg.de/public/papers/hbam/hbam2011-07.pdf}., (): 2011. Google Scholar |
[29] |
M. D. Gunzburger, "Perspectives in Flow Control and Optimization,", Advances in Design and Control, 5 (2003).
|
[30] |
M. D. Gunzburger, L. S. Hou and Th. P. Svobodny, Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls,, RAIRO Modél. Math. Anal. Numér., 25 (1991), 711.
|
[31] |
M. D. Gunzburger and S. Manservisi, Analysis and approximation of the velocity tracking problem for Navier-Stokes flows with distributed control,, SIAM J. Numer. Anal., 37 (2000), 1481.
doi: 10.1137/S0036142997329414. |
[32] |
F. Hecht, FreeFem++,, Third edition, (2011). Google Scholar |
[33] |
M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case,, Comput. Optim. Appl., 30 (2005), 45.
doi: 10.1007/s10589-005-4559-5. |
[34] |
M. Hinze and K. Kunisch, Second order methods for optimal control of time-dependent fluid flow,, SIAM J. Control and Optim., 40 (2001), 925.
doi: 10.1137/S0363012999361810. |
[35] |
K. Ito and K. Kunisch, "Lagrange Multiplier Approach to Variational Problems and Applications,", Advances in Design and Control, 15 (2008).
|
[36] |
G. Knowles, Finite element approximation of parabolic time optimal control problems,, SIAM J. Control and Optim., 20 (1982), 414.
doi: 10.1137/0320032. |
[37] |
I. Lasiecka, Rietz-Galerkin approximation of the time optimal boundary control problem for parabolic systems with Dirichlet boundary conditions,, SIAM J. Control and Optim., 22 (1984), 477.
|
[38] |
I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations,", Cambridge University Press, (2000). Google Scholar |
[39] |
J.-L. Lions, "Some Aspects of the Control of Distributed Parameter Systems,", Conference Board of the Mathematical Sciences, (1972). Google Scholar |
[40] |
W.-B. Liu and N. Yan, A posteriori error estimates for optimal control problems governed by parabolic equations,, Numer. Math., 93 (2003), 497.
doi: 10.1007/s002110100380. |
[41] |
W.-B. Liu, H.-P. 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.
doi: 10.1137/S0036142902397090. |
[42] |
Y. Liu, R. E. Bank, T. F. Dupont, S. Garcia and R. F. Santos, Symmetric error estimates for moving mesh mixed methods for advection-diffusion equations,, SIAM J. Numer. Anal., 40 (2002), 2270.
doi: 10.1137/S003614290038073X. |
[43] |
K. Malanowski, Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal-control problems,, Appl. Math. Optim., 8 (1982), 69.
doi: 10.1007/BF01447752. |
[44] |
D. Meidner and B. Vexler, Adaptive space-time finite element methods for parabolic optimization problems,, SIAM J. Control and Optim., 46 (2007), 116.
doi: 10.1137/060648994. |
[45] |
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 and Optim., 47 (2008), 1150.
doi: 10.1137/070694016. |
[46] |
D. Meidner and B. Vexler, A-priori error analysis of the Petrov-Galerkin Crank-Nicolson scheme for parabolic optimal control problems,, SIAM J. Control and Optim., 49 (2011), 2183.
doi: 10.1137/100809611. |
[47] |
P. Neittaanmäki and D. Tiba, "Optimal Control of Nonlinear Parabolic Systems. Theory, Algorithms and Applications,", Monographs and Textbooks in Pure and Applied Mathematics, 179 (1994).
|
[48] |
I. Neitzel and B. Vexler, A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems,, Numer. Math., 120 (2011), 345.
doi: 10.1007/s00211-011-0409-9. |
[49] |
A. Rösch, Error estimates for parabolic optimal control problems with control constraints,, Zeitschrift für Analysis und ihre Anwendunges, 23 (2004), 353.
|
[50] |
V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems,", Springer Series in Computational Mathematics, 25 (1997).
|
[51] |
F. Tröltzsch, Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems,, in, 111 (1993), 57.
|
[52] |
F. Tröltzsch, Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems-strong convergence of optimal controls,, Appl. Math. Optim., 29 (1994), 309.
|
[53] |
F. Tröltzsch, "Optimal Control of Partial Differential Equations: Theory, Methods and Applications,", Graduate Studies in Mathematics, 112 (2010).
|
[54] |
N. J. Walkington, Compactness properties of the DG and CG time stepping schemes for parabolic equations,, SIAM J. Numer. Anal, 47 (2010), 4680.
doi: 10.1137/080728378. |
[55] |
R. Winther, Error estimates for a Galerkin approximation of a parabolic control problem,, Ann. Math. Pura Appl. (4), 117 (1978), 173.
doi: 10.1007/BF02417890. |
[56] |
R. Winther, Initial value methods for parabolic control problems,, Math. Comp., 34 (1980), 115.
doi: 10.1090/S0025-5718-1980-0551293-7. |
[57] |
E. Zeidler, "Nonlinear Functional Analysis and its Applications. II/B. Nonlinear Monotone Operators,", Springer-Verlag, (1990).
doi: 10.1007/978-1-4612-0985-0. |
show all references
References:
[1] |
G. Akrivis and C. Makridakis, Galerkin time-stepping methods for nonlinear parabolic equations,, M2AN Math. Model. and Numer. Anal., 38 (2004), 261.
doi: 10.1051/m2an:2004013. |
[2] |
G. Allaire and O. Pantz, Structural optimization with FreeFem++,, Struct. Multidiscip. Optim., 32 (2006), 173.
doi: 10.1007/s00158-006-0017-y. |
[3] |
T. Apel and T. Flaig, Crank-Nicolson schemes for optimal control problems with evolution equations,, submitted. Available from: \url{http://www.unibw.de/bauv1/personen/apel/papers}., (). Google Scholar |
[4] |
A. Borzi and R. Griesse, Distributed optimal control for lambda-omega systems,, J. Numer. Math., 14 (2006), 17.
doi: 10.1515/156939506776382120. |
[5] |
E. Casas and J.-P. Raymond, Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equation,, SIAM J. Control and Optim., 45 (2006), 1586.
doi: 10.1137/050626600. |
[6] |
E. Casas, M. Mateos and F. Tröltzsch, Error estimates for the numerical approximation of boundary semilinear elliptic control problem,, Comput. Optim. and Appl., 31 (2005), 193.
|
[7] |
K. Chrysafinos, Discontinous Galerkin approximations for distributed optimal control problems constrained by parabolic PDE's,, Int. J. Numer. Anal. and Mod., 4 (2007), 690.
|
[8] |
K. Chrysafinos, Analysis and finite element approximations for distributed optimal control problems for implicit parabolic equations,, J. Comput. Appl. Math., 231 (2009), 327.
doi: 10.1016/j.cam.2009.02.092. |
[9] |
K. Chrysafinos, Convergence of discontinuous Galerkin approximations of an optimal control problem associated to semilinear parabolic PDE's,, M2AN Math. Model. Numer. Anal., 44 (2010), 189.
|
[10] |
K. Chrysafinos, Convergence of discontinuous time-stepping schemes for a Robin boundary control problem under minimal regularity assumptions,, submitted. Available from: \url{http://www.math.ntua.gr/~chrysafinos}., (). Google Scholar |
[11] |
K. Chrysafinos and N. J. Walkington, Error estimates for the discontinuous Galerkin methods for parabolic equations,, SIAM J. Numer. Anal., 44 (2006), 349.
doi: 10.1137/030602289. |
[12] |
K. Chrysafinos and N. J. Walkington, Discontinuous Galerkin approximations of the Stokes and Navier-Stokes equations,, Math. Comp., 79 (2010), 2135.
doi: 10.1090/S0025-5718-10-02348-3. |
[13] |
K. Chrysafinos and N. J. Walkington, Lagrangian and moving mesh methods for the convection diffusion equation,, M2AN Math. Model. Numer. Anal., 42 (2008), 25.
|
[14] |
K. Chrysafinos, M. D. Gunzburger and L. S. Hou, Semidiscrete approximations of optimal Robin boundary control problems constrained by semilinear parabolic PDE,, J. Math. Anal. Appl., 323 (2006), 891.
doi: 10.1016/j.jmaa.2005.10.053. |
[15] |
P. G. Ciarlet, "The Finite Element Method for Elliptic Problems,", Reprint of the 1978 original, (1978).
|
[16] |
K. Deckelnick and M. Hinze, Semidiscretization and error estimates for distributed control of the instationary Navier-Stokes equations,, Numer. Math., 97 (2004), 297.
doi: 10.1007/s00211-003-0507-4. |
[17] |
K. Deckelnick and M. Hinze, Variational discretization of parabolic control problems in the presence of pointwise state constraints,, J. Comput. Math., 29 (2011), 1.
|
[18] |
T. F. Dupont and Y. Liu, Symmetric error estimates for moving mesh Galerkin methods for advection-diffusion equations,, SIAM J. Numer. Anal., 40 (2002), 914.
doi: 10.1137/S0036142900380431. |
[19] |
D. Estep and S. Larsson, The discontinuous Galerkin method for semilinear parabolic problems,, RAIRO Modél. Math. Anal. Numér., 27 (1993), 35.
|
[20] |
K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem,, SIAM J. Numer. Anal., 28 (1991), 43.
doi: 10.1137/0728003. |
[21] |
K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. II. Optimal error estimates in $L_{\infty}(L^2)$ and $L_{\infty}(L_{\infty})$,, SIAM J. Numer. Anal., 32 (1995), 706.
doi: 10.1137/0732033. |
[22] |
K. Ericksson and C. Johnson, Adaptive finite element methods for parabolic problems IV, Nonlinear problems,, SIAM J. Numer. Anal., 32 (1995), 1729.
doi: 10.1137/0732078. |
[23] |
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.
|
[24] |
L. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (1998).
|
[25] |
R. Falk, Approximation of a class of otimal control problems with order of convergence estimates,, J. Math. Anal. Appl., 44 (1973), 28.
doi: 10.1016/0022-247X(73)90022-X. |
[26] |
A. Fursikov, "Optimal Control of Distributed Systems. Theory and Applications,", Translations of Mathematical Monographs, 187 (2000).
|
[27] |
V. Girault and P.-A. Raviart, "Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms,", Springer Series in Computational Mathematics, 5 (1986).
|
[28] |
W. Gong, M. Hinze and Z. Zhou, A priori error analysis for finite element approximation of parabolic optimal control problems with pointwise control,, submitted. Available from: \url{http://preprint.math.uni-hamburg.de/public/papers/hbam/hbam2011-07.pdf}., (): 2011. Google Scholar |
[29] |
M. D. Gunzburger, "Perspectives in Flow Control and Optimization,", Advances in Design and Control, 5 (2003).
|
[30] |
M. D. Gunzburger, L. S. Hou and Th. P. Svobodny, Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls,, RAIRO Modél. Math. Anal. Numér., 25 (1991), 711.
|
[31] |
M. D. Gunzburger and S. Manservisi, Analysis and approximation of the velocity tracking problem for Navier-Stokes flows with distributed control,, SIAM J. Numer. Anal., 37 (2000), 1481.
doi: 10.1137/S0036142997329414. |
[32] |
F. Hecht, FreeFem++,, Third edition, (2011). Google Scholar |
[33] |
M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case,, Comput. Optim. Appl., 30 (2005), 45.
doi: 10.1007/s10589-005-4559-5. |
[34] |
M. Hinze and K. Kunisch, Second order methods for optimal control of time-dependent fluid flow,, SIAM J. Control and Optim., 40 (2001), 925.
doi: 10.1137/S0363012999361810. |
[35] |
K. Ito and K. Kunisch, "Lagrange Multiplier Approach to Variational Problems and Applications,", Advances in Design and Control, 15 (2008).
|
[36] |
G. Knowles, Finite element approximation of parabolic time optimal control problems,, SIAM J. Control and Optim., 20 (1982), 414.
doi: 10.1137/0320032. |
[37] |
I. Lasiecka, Rietz-Galerkin approximation of the time optimal boundary control problem for parabolic systems with Dirichlet boundary conditions,, SIAM J. Control and Optim., 22 (1984), 477.
|
[38] |
I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations,", Cambridge University Press, (2000). Google Scholar |
[39] |
J.-L. Lions, "Some Aspects of the Control of Distributed Parameter Systems,", Conference Board of the Mathematical Sciences, (1972). Google Scholar |
[40] |
W.-B. Liu and N. Yan, A posteriori error estimates for optimal control problems governed by parabolic equations,, Numer. Math., 93 (2003), 497.
doi: 10.1007/s002110100380. |
[41] |
W.-B. Liu, H.-P. 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.
doi: 10.1137/S0036142902397090. |
[42] |
Y. Liu, R. E. Bank, T. F. Dupont, S. Garcia and R. F. Santos, Symmetric error estimates for moving mesh mixed methods for advection-diffusion equations,, SIAM J. Numer. Anal., 40 (2002), 2270.
doi: 10.1137/S003614290038073X. |
[43] |
K. Malanowski, Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal-control problems,, Appl. Math. Optim., 8 (1982), 69.
doi: 10.1007/BF01447752. |
[44] |
D. Meidner and B. Vexler, Adaptive space-time finite element methods for parabolic optimization problems,, SIAM J. Control and Optim., 46 (2007), 116.
doi: 10.1137/060648994. |
[45] |
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 and Optim., 47 (2008), 1150.
doi: 10.1137/070694016. |
[46] |
D. Meidner and B. Vexler, A-priori error analysis of the Petrov-Galerkin Crank-Nicolson scheme for parabolic optimal control problems,, SIAM J. Control and Optim., 49 (2011), 2183.
doi: 10.1137/100809611. |
[47] |
P. Neittaanmäki and D. Tiba, "Optimal Control of Nonlinear Parabolic Systems. Theory, Algorithms and Applications,", Monographs and Textbooks in Pure and Applied Mathematics, 179 (1994).
|
[48] |
I. Neitzel and B. Vexler, A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems,, Numer. Math., 120 (2011), 345.
doi: 10.1007/s00211-011-0409-9. |
[49] |
A. Rösch, Error estimates for parabolic optimal control problems with control constraints,, Zeitschrift für Analysis und ihre Anwendunges, 23 (2004), 353.
|
[50] |
V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems,", Springer Series in Computational Mathematics, 25 (1997).
|
[51] |
F. Tröltzsch, Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems,, in, 111 (1993), 57.
|
[52] |
F. Tröltzsch, Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems-strong convergence of optimal controls,, Appl. Math. Optim., 29 (1994), 309.
|
[53] |
F. Tröltzsch, "Optimal Control of Partial Differential Equations: Theory, Methods and Applications,", Graduate Studies in Mathematics, 112 (2010).
|
[54] |
N. J. Walkington, Compactness properties of the DG and CG time stepping schemes for parabolic equations,, SIAM J. Numer. Anal, 47 (2010), 4680.
doi: 10.1137/080728378. |
[55] |
R. Winther, Error estimates for a Galerkin approximation of a parabolic control problem,, Ann. Math. Pura Appl. (4), 117 (1978), 173.
doi: 10.1007/BF02417890. |
[56] |
R. Winther, Initial value methods for parabolic control problems,, Math. Comp., 34 (1980), 115.
doi: 10.1090/S0025-5718-1980-0551293-7. |
[57] |
E. Zeidler, "Nonlinear Functional Analysis and its Applications. II/B. Nonlinear Monotone Operators,", Springer-Verlag, (1990).
doi: 10.1007/978-1-4612-0985-0. |
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