Error estimates of variational discretization for semilinear parabolic optimal control problems. (English) Zbl 1484.49060
Summary: In this paper, variational discretization directed against the optimal control problem governed by nonlinear parabolic equations with control constraints is studied. It is known that the a priori error estimates is \(|||u-u_h|||_{L^\infty(J; L^2(\Omega))} = O(h+k)\) using backward Euler method for standard finite element. In this paper, the better result \(|||u-u_h|||_{L^\infty(J; L^2(\Omega))} = O(h^2+k)\) is gained. Beyond that, we get a posteriori error estimates of residual type.
MSC:
| 49M41 | PDE constrained optimization (numerical aspects) |
| 49M25 | Discrete approximations in optimal control |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
optimal control problems; semilinear parabolic equations; error estimates; finite element methodsReferences:
| [1] | P. Neittaanmaki, D. Tiba, <i>Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms</i> <i>and Applications</i>, Dekker, New Nork, 1994. · Zbl 0812.49001 |
| [2] | D. Tiba, <i>Lectures on The Optimal Control of Elliptic Equations</i>, University of Jyvaskyla Press, Finland, 1995. |
| [3] | W. Liu, N. Yan, <i>Adaptive Finite Element Methods For Optimal Control Governed by PDEs</i>, Science Press, Beijing, 2008. |
| [4] | Y. Chen, Z. Lu, W. Liu, <i>Numerical solution of partial differential equation</i>, Science Press, Beijing, 2015. |
| [5] | T, Error estimates and superconvergence of a mixed finite element method for elliptic optimal control problems, Comput. Math. Appl., 74, 714-726 (2017) · Zbl 1385.49021 · doi:10.1016/j.camwa.2017.05.021 |
| [6] | P. Ciarlet, <i>The Finite Element Method for Elliptic Problems</i>, North-Holland, Amstterdam, 1978. · Zbl 0383.65058 |
| [7] | Y, A legendre galerkin spectral method for optimal control problems governed by elliptic equations, SIAM J. Numer. Anal., 46, 2254-2275 (2008) · Zbl 1175.49003 · doi:10.1137/070679703 |
| [8] | Y, Superconvergence for optimal control problems gonverned by semi-linear elliptic equations, J. Sci. Comput., 39, 206-221 (2009) · Zbl 1203.65177 · doi:10.1007/s10915-008-9258-9 |
| [9] | X, <i>L</i><sup>∞;</sup>-error estimates for general optimal control problem by mixed finite element methods, Int. J. Numer. Anal. Model., 5, 441-456 (2008) · Zbl 1160.35551 |
| [10] | R, Adaptive finite element approximation for distributed elliptic optimal control problems, SIAM J. Control Optim., 41, 1321-1349 (2002) · Zbl 1034.49031 · doi:10.1137/S0363012901389342 |
| [11] | P, Optimal control of partial differential equations, SIAM J. Control Optim., 50, 943-963 (2012) · Zbl 1244.49038 · doi:10.1137/100815037 |
| [12] | W, A posteriori error estimates for optimal control problems governed by parabolic equations, Numer. Math., 93, 497-521 (2003) · Zbl 1049.65057 · doi:10.1007/s002110100380 |
| [13] | B. I. Ananyev, <i>A control problem for parabolic systems with incomplete information</i>, Mathematical Optimization Theory and Operations Research, Springer, Cham, 2019. · Zbl 1439.49063 |
| [14] | H, Recovery type superconvergence and a posteriori error estimates for control problems governed by Stokes equations, J. Comput. Appl. Math., 209, 187-207 (2007) · Zbl 1140.65053 · doi:10.1016/j.cam.2006.10.083 |
| [15] | W, A posteriori error estimates for control problems governed by Stokes equations, SIAM J. Numer. Anal., 40, 1850-1869 (2002) · Zbl 1028.49025 · doi:10.1137/S0036142901384009 |
| [16] | J. Lions, <i>Optimal Control of Systems Governed by Partial Differential Equations</i>, Springer-Verlag, Berlin, 1971. · Zbl 0203.09001 |
| [17] | J. Lions, E. Magenes, <i>Non Homogeneous Boundary Value Problems and Applications</i>, Springer-verlag, Berlin, 1972. · Zbl 0223.35039 |
| [18] | H, A priori error estimates for optimal problems governed by transient advection-diffusion equations, J. Sci. Comput., 38, 290-315 (2009) · Zbl 1203.65163 · doi:10.1007/s10915-008-9224-6 |
| [19] | W, A posteriori error estimates for distributed convex optimal control problems, Adv. Comput. Math., 15, 285-309 (2001) · Zbl 1008.49024 · doi:10.1023/A:1014239012739 |
| [20] | R, A posteriori error estimates of recovery type for distributed convex optimal control problems, J. Sci. Comput., 33, 155-182 (2007) · Zbl 1128.65048 · doi:10.1007/s10915-007-9147-7 |
| [21] | N, postriori error estimators of gradient recovery type for FEM of a model optimal control problem, Adv. Comp. Math., 19, 323-336 (2003) · Zbl 1027.65087 · doi:10.1023/A:1022800401298 |
| [22] | Y, Variational discretization for parabolic optimal control problems with control constraints, J. Systems Sci. Compl., 25, 880-895 (2012) · Zbl 1269.49054 · doi:10.1007/s11424-012-0279-y |
| [23] | M, A variational discretization concept in cotrol constrained optimization: the linear-quadratic case, Comput. Optim. Appl., 30, 45-63 (2005) · Zbl 1074.65069 · doi:10.1007/s10589-005-4559-5 |
| [24] | M, Variational discretization for optimal control governed by convection dominated diffusion equations, J. Comput. Math., 27, 237-253 (2009) · Zbl 1212.65248 |
| [25] | M. Huang, <i>Numerical Methods for Evelution Equations</i>, Secience Press, Beijing, 2004. |
| [26] | A. Kufner, O. John, S. Fuck, <i>Function Spaces</i>, Nordhoff, Leiden, The Netherlands, 1997. |
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