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A posteriori \(L^\infty(L^\infty)\)-error estimates for finite-element approximations to parabolic optimal control problems. (English) Zbl 07453248

Summary: We derive space-time a posteriori error estimates of finite-element method for the linear parabolic optimal control problems in a convex bounded polyhedral domain. The variational discretization is used to approximate the state and co-state variables with the piecewise linear and continuous functions, while the control variable is computed using the implicit relation between the control and co-state variables. The temporal discretization is based on the backward Euler method. The key feature of this approach is not to discretize the control variable but to implicitly utilize the optimality conditions for the discretization of the control variable. Our error analysis relies on the elliptic reconstruction technique introduced by C. Makridakis and R. H. Nochetto [SIAM J. Numer. Anal. 41, No. 4, 1585–1594 (2003; Zbl 1052.65088)] in conjunction with heat kernel estimates for linear parabolic problem. The use of elliptic reconstruction technique greatly simplifies the analysis by allowing us to take the advantage of existing elliptic maximum norm error estimate and the heat kernel estimate. We derive a posteriori error estimates for the state, co-state, and control variables in the \(L^\infty (0,T;\, L^\infty (\varOmega))\)-norm. Numerical experiments are conducted to illustrate the performance of the derived estimators.

MSC:

49J20 Existence theories for optimal control problems involving partial differential equations
49M05 Numerical methods based on necessary conditions
49M15 Newton-type methods
49M25 Discrete approximations in optimal control
49M29 Numerical methods involving duality
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Citations:

Zbl 1052.65088
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References:

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