A posteriori L∞(L∞)-error estimates for finite-element approximations to parabolic optimal control problems

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 function...

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Vydané v:Computational & applied mathematics Ročník 40; číslo 8
Hlavní autori: Manohar, Ram, Sinha, Rajen Kumar
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Cham Springer International Publishing 01.12.2021
Springer Nature B.V
Predmet:
Applications of Mathematics
Applied physics
Computational mathematics
Computational Mathematics and Numerical Analysis
Continuity (mathematics)
Discretization
Error analysis
Estimates
Finite element method
Kernels
Mathematical Applications in Computer Science
Mathematical Applications in the Physical Sciences
Mathematics
Mathematics and Statistics
Optimal control
Optimization
Reconstruction
State variable
Variables
Variational discretization
Elliptic reconstruction
49J20
Parabolic optimal control problem
Maximum norm error estimates
49M29
A posteriori error estimates
49M05
49M15
49M25
Backward-Euler scheme
65N30
ISSN:2238-3603, 1807-0302
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Popis
Shrnutí: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 Makridakis and Nochetto (SIAM J Numer Anal, 41:1585–1594, 2003) 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 ∞ ( 0 , T ; L ∞ ( Ω ) ) -norm. Numerical experiments are conducted to illustrate the performance of the derived estimators.
Bibliografia:ObjectType-Article-1
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content type line 14
ISSN:2238-3603
1807-0302
DOI:10.1007/s40314-021-01682-5