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Space‐time a posteriori error analysis of finite element approximation for parabolic optimal control problems: A reconstruction approach
Author(s) -
Manohar Ram,
Kumar Sinha Rajen
Publication year - 2020
Publication title -
optimal control applications and methods
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.458
H-Index - 44
eISSN - 1099-1514
pISSN - 0143-2087
DOI - 10.1002/oca.2618
Subject(s) - mathematics , discretization , finite element method , piecewise , a priori and a posteriori , backward euler method , piecewise linear function , norm (philosophy) , estimator , optimal control , bounded function , mathematical analysis , mathematical optimization , statistics , physics , political science , law , thermodynamics , philosophy , epistemology
Summary We derive space‐time a posteriori error estimates of finite element method for linear parabolic optimal control problems in a bounded convex polygonal domain. To discretize the control problem, we use piecewise linear and continuous finite elements for the approximations of the state and costate variables, whereas piecewise constant functions are employed for the control variable. The temporal discretization is based on the backward Euler implicit scheme. An elliptic reconstruction technique in conjunction with energy argument is used to derive a posteriori error estimates for the state, costate, and control variables in the L ∞ (0, T ; L 2 (Ω))‐norm. Moreover, numerical experiments are performed to illustrate the performance of the derived estimators.