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Volume 33, Issue 2
Mixed Discontinuous Galerkin Time-Stepping Method for Linear Parabolic Optimal Control Problems

Tianliang Hou & Yanping Chen

DOI: 10.4208/jcm.1211-m4267

J. Comp. Math., 33 (2015), pp. 158-178.

Published online: 2015-04

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  • Abstract

In this paper, we discuss the mixed discontinuous Galerkin (DG) finite element approximation to linear parabolic optimal control problems. For the state variables and the co-state variables, the discontinuous finite element method is used for the time discretization and the Raviart-Thomas mixed finite element method is used for the space discretization. We do not discretize the space of admissible control but implicitly utilize the relation between co-state and control for the discretization of the control. We derive a priori error estimates for the lowest order mixed DG finite element approximation. Moveover, for the element of arbitrary order in space and time, we derive a posteriori $L^2(0, T ;L^2(Ω))$ error estimates for the scalar functions, assuming that only the underlying mesh is static. Finally, we present an example to confirm the theoretical result on a priori error estimates.

  • Keywords

A priori error estimates, A posteriori error estimates, Mixed finite element, Discontinuous Galerkin method, Parabolic control problems.

  • AMS Subject Headings

35K10, 65N30.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

htlchb@163.com (Tianliang Hou)

yanpingchen@scnu.edu.cn (Yanping Chen)

  • BibTex
  • RIS
  • TXT
@Article{JCM-33-158, author = {Hou , Tianliang and Chen , Yanping}, title = {Mixed Discontinuous Galerkin Time-Stepping Method for Linear Parabolic Optimal Control Problems}, journal = {Journal of Computational Mathematics}, year = {2015}, volume = {33}, number = {2}, pages = {158--178}, abstract = {

In this paper, we discuss the mixed discontinuous Galerkin (DG) finite element approximation to linear parabolic optimal control problems. For the state variables and the co-state variables, the discontinuous finite element method is used for the time discretization and the Raviart-Thomas mixed finite element method is used for the space discretization. We do not discretize the space of admissible control but implicitly utilize the relation between co-state and control for the discretization of the control. We derive a priori error estimates for the lowest order mixed DG finite element approximation. Moveover, for the element of arbitrary order in space and time, we derive a posteriori $L^2(0, T ;L^2(Ω))$ error estimates for the scalar functions, assuming that only the underlying mesh is static. Finally, we present an example to confirm the theoretical result on a priori error estimates.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1211-m4267}, url = {http://global-sci.org/intro/article_detail/jcm/9834.html} }
TY - JOUR T1 - Mixed Discontinuous Galerkin Time-Stepping Method for Linear Parabolic Optimal Control Problems AU - Hou , Tianliang AU - Chen , Yanping JO - Journal of Computational Mathematics VL - 2 SP - 158 EP - 178 PY - 2015 DA - 2015/04 SN - 33 DO - http://doi.org/10.4208/jcm.1211-m4267 UR - https://global-sci.org/intro/article_detail/jcm/9834.html KW - A priori error estimates, A posteriori error estimates, Mixed finite element, Discontinuous Galerkin method, Parabolic control problems. AB -

In this paper, we discuss the mixed discontinuous Galerkin (DG) finite element approximation to linear parabolic optimal control problems. For the state variables and the co-state variables, the discontinuous finite element method is used for the time discretization and the Raviart-Thomas mixed finite element method is used for the space discretization. We do not discretize the space of admissible control but implicitly utilize the relation between co-state and control for the discretization of the control. We derive a priori error estimates for the lowest order mixed DG finite element approximation. Moveover, for the element of arbitrary order in space and time, we derive a posteriori $L^2(0, T ;L^2(Ω))$ error estimates for the scalar functions, assuming that only the underlying mesh is static. Finally, we present an example to confirm the theoretical result on a priori error estimates.

Hou , Tianliang and Chen , Yanping. (2015). Mixed Discontinuous Galerkin Time-Stepping Method for Linear Parabolic Optimal Control Problems. Journal of Computational Mathematics. 33 (2). 158-178. doi:10.4208/jcm.1211-m4267
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