Volume 12, Issue 1
Fully Discrete $H$1-Galerkin Mixed Finite Element Methods for Parabolic Optimal Control Problems

Numer. Math. Theor. Meth. Appl., 12 (2019), pp. 134-153.

Published online: 2018-09

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• Abstract

In this paper, we investigate a priori and a posteriori error estimates of fully discrete $H$1-Galerkin mixed finite element methods for parabolic optimal control problems. The state variables and co-state variables are approximated by the lowest order Raviart-Thomas mixed finite element and linear finite element, and the control variable is approximated by piecewise constant functions. The time discretization of the state and co-state are based on finite difference methods. First, we derive a priori error estimates for the control variable, the state variables and the adjoint state variables. Second, by use of energy approach, we derive a posteriori error estimates for optimal control problems, assuming that only the underlying mesh is static. A numerical example is presented to verify the theoretical results on a priori error estimates.

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@Article{NMTMA-12-134, author = {}, title = {Fully Discrete $H$1-Galerkin Mixed Finite Element Methods for Parabolic Optimal Control Problems}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2018}, volume = {12}, number = {1}, pages = {134--153}, abstract = {

In this paper, we investigate a priori and a posteriori error estimates of fully discrete $H$1-Galerkin mixed finite element methods for parabolic optimal control problems. The state variables and co-state variables are approximated by the lowest order Raviart-Thomas mixed finite element and linear finite element, and the control variable is approximated by piecewise constant functions. The time discretization of the state and co-state are based on finite difference methods. First, we derive a priori error estimates for the control variable, the state variables and the adjoint state variables. Second, by use of energy approach, we derive a posteriori error estimates for optimal control problems, assuming that only the underlying mesh is static. A numerical example is presented to verify the theoretical results on a priori error estimates.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2019.m1623}, url = {http://global-sci.org/intro/article_detail/nmtma/12694.html} }
TY - JOUR T1 - Fully Discrete $H$1-Galerkin Mixed Finite Element Methods for Parabolic Optimal Control Problems JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 134 EP - 153 PY - 2018 DA - 2018/09 SN - 12 DO - http://doi.org/10.4208/nmtma.2019.m1623 UR - https://global-sci.org/intro/article_detail/nmtma/12694.html KW - AB -

In this paper, we investigate a priori and a posteriori error estimates of fully discrete $H$1-Galerkin mixed finite element methods for parabolic optimal control problems. The state variables and co-state variables are approximated by the lowest order Raviart-Thomas mixed finite element and linear finite element, and the control variable is approximated by piecewise constant functions. The time discretization of the state and co-state are based on finite difference methods. First, we derive a priori error estimates for the control variable, the state variables and the adjoint state variables. Second, by use of energy approach, we derive a posteriori error estimates for optimal control problems, assuming that only the underlying mesh is static. A numerical example is presented to verify the theoretical results on a priori error estimates.

Tianliang Hou, Chunmei Liu & Hongbo Chen. (2020). Fully Discrete $H$1-Galerkin Mixed Finite Element Methods for Parabolic Optimal Control Problems. Numerical Mathematics: Theory, Methods and Applications. 12 (1). 134-153. doi:10.4208/nmtma.2019.m1623
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