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Volume 40, Issue 2
Elliptic Reconstruction and a Posteriori Error Estimates for Fully Discrete Semilinear Parabolic Optimal Control Problems

Ram Manohar & Rajen Kumar Sinha

DOI: 10.4208/jcm.2009-m2019-0194

J. Comp. Math., 40 (2022), pp. 147-176.

Published online: 2022-01

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  • Abstract

This article studies a posteriori error analysis of fully discrete finite element approximations for semilinear parabolic optimal control problems. Based on elliptic reconstruction approach introduced earlier by Makridakis and Nochetto [25], a residual based a posteriori error estimators for the state, co-state and control variables are derived. The space discretization of the state and co-state variables is done by using the piecewise linear and continuous finite elements, whereas the piecewise constant functions are employed for the control variable. The temporal discretization is based on the backward Euler method. We derive a posteriori error estimates for the state, co-state and control variables in the $L^\infty(0,T;L^2(\Omega))$-norm. Finally, a numerical experiment  is performed to illustrate the performance of the derived estimators.

  • Keywords

Semilinear parabolic optimal control problem, Finite element method, The backward Euler method, Elliptic reconstruction, A posteriori error estimates.

  • AMS Subject Headings

49J20, 65J15, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

rmanohar267@gmail.com (Ram Manohar)

rajen@iitg.ac.in (Rajen Kumar Sinha)

  • BibTex
  • RIS
  • TXT
@Article{JCM-40-147, author = {Manohar , Ram and Kumar Sinha , Rajen}, title = {Elliptic Reconstruction and a Posteriori Error Estimates for Fully Discrete Semilinear Parabolic Optimal Control Problems}, journal = {Journal of Computational Mathematics}, year = {2022}, volume = {40}, number = {2}, pages = {147--176}, abstract = {

This article studies a posteriori error analysis of fully discrete finite element approximations for semilinear parabolic optimal control problems. Based on elliptic reconstruction approach introduced earlier by Makridakis and Nochetto [25], a residual based a posteriori error estimators for the state, co-state and control variables are derived. The space discretization of the state and co-state variables is done by using the piecewise linear and continuous finite elements, whereas the piecewise constant functions are employed for the control variable. The temporal discretization is based on the backward Euler method. We derive a posteriori error estimates for the state, co-state and control variables in the $L^\infty(0,T;L^2(\Omega))$-norm. Finally, a numerical experiment  is performed to illustrate the performance of the derived estimators.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2009-m2019-0194}, url = {http://global-sci.org/intro/article_detail/jcm/20182.html} }
TY - JOUR T1 - Elliptic Reconstruction and a Posteriori Error Estimates for Fully Discrete Semilinear Parabolic Optimal Control Problems AU - Manohar , Ram AU - Kumar Sinha , Rajen JO - Journal of Computational Mathematics VL - 2 SP - 147 EP - 176 PY - 2022 DA - 2022/01 SN - 40 DO - http://doi.org/10.4208/jcm.2009-m2019-0194 UR - https://global-sci.org/intro/article_detail/jcm/20182.html KW - Semilinear parabolic optimal control problem, Finite element method, The backward Euler method, Elliptic reconstruction, A posteriori error estimates. AB -

This article studies a posteriori error analysis of fully discrete finite element approximations for semilinear parabolic optimal control problems. Based on elliptic reconstruction approach introduced earlier by Makridakis and Nochetto [25], a residual based a posteriori error estimators for the state, co-state and control variables are derived. The space discretization of the state and co-state variables is done by using the piecewise linear and continuous finite elements, whereas the piecewise constant functions are employed for the control variable. The temporal discretization is based on the backward Euler method. We derive a posteriori error estimates for the state, co-state and control variables in the $L^\infty(0,T;L^2(\Omega))$-norm. Finally, a numerical experiment  is performed to illustrate the performance of the derived estimators.

Manohar , Ram and Kumar Sinha , Rajen. (2022). Elliptic Reconstruction and a Posteriori Error Estimates for Fully Discrete Semilinear Parabolic Optimal Control Problems. Journal of Computational Mathematics. 40 (2). 147-176. doi:10.4208/jcm.2009-m2019-0194
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