Volume 13, Issue 2
Decoupled Mixed Element Methods for Fourth Order Elliptic Optimal Control Problems with Control Constraints

Numer. Math. Theor. Meth. Appl., 13 (2020), pp. 400-432.

Published online: 2020-03

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• Abstract

In this paper, we study the finite element methods for distributed optimal control problems governed by the biharmonic operator. Motivated from reducing the regularity of solution space, we use the decoupled mixed element method which was used to approximate the solution of biharmonic equation to solve the fourth order optimal control problems. Two finite element schemes, i.e., Lagrange conforming element combined with full control discretization and the nonconforming Crouzeix-Raviart element combined with variational control discretization, are used to discretize the decoupled optimal control system. The corresponding a priori error estimates are derived under appropriate norms which are then verified by extensive numerical experiments.

• Keywords

Fourth order elliptic equation, optimal control problem, decoupled mixed element method, Lagrange element, nonconforming Crouzeix-Raviart element, a priori error estimates.

49J20, 35J35, 65K10, 65N15, 65N30

yshen@xauat.edu.cn (Yue Shen)

787479522@qq.com (Chang Jin)

• BibTex
• RIS
• TXT
@Article{NMTMA-13-400, author = {Shen , Yue and Jin , Chang}, title = {Decoupled Mixed Element Methods for Fourth Order Elliptic Optimal Control Problems with Control Constraints}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2020}, volume = {13}, number = {2}, pages = {400--432}, abstract = {

In this paper, we study the finite element methods for distributed optimal control problems governed by the biharmonic operator. Motivated from reducing the regularity of solution space, we use the decoupled mixed element method which was used to approximate the solution of biharmonic equation to solve the fourth order optimal control problems. Two finite element schemes, i.e., Lagrange conforming element combined with full control discretization and the nonconforming Crouzeix-Raviart element combined with variational control discretization, are used to discretize the decoupled optimal control system. The corresponding a priori error estimates are derived under appropriate norms which are then verified by extensive numerical experiments.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2019-0016}, url = {http://global-sci.org/intro/article_detail/nmtma/15468.html} }
TY - JOUR T1 - Decoupled Mixed Element Methods for Fourth Order Elliptic Optimal Control Problems with Control Constraints AU - Shen , Yue AU - Jin , Chang JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 400 EP - 432 PY - 2020 DA - 2020/03 SN - 13 DO - http://doi.org/10.4208/nmtma.OA-2019-0016 UR - https://global-sci.org/intro/article_detail/nmtma/15468.html KW - Fourth order elliptic equation, optimal control problem, decoupled mixed element method, Lagrange element, nonconforming Crouzeix-Raviart element, a priori error estimates. AB -

In this paper, we study the finite element methods for distributed optimal control problems governed by the biharmonic operator. Motivated from reducing the regularity of solution space, we use the decoupled mixed element method which was used to approximate the solution of biharmonic equation to solve the fourth order optimal control problems. Two finite element schemes, i.e., Lagrange conforming element combined with full control discretization and the nonconforming Crouzeix-Raviart element combined with variational control discretization, are used to discretize the decoupled optimal control system. The corresponding a priori error estimates are derived under appropriate norms which are then verified by extensive numerical experiments.

Yue Shen & Chang Jin. (2020). Decoupled Mixed Element Methods for Fourth Order Elliptic Optimal Control Problems with Control Constraints. Numerical Mathematics: Theory, Methods and Applications. 13 (2). 400-432. doi:10.4208/nmtma.OA-2019-0016
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