Volume 13, Issue 4
A Splitting Least-Squares Mixed Finite Element Method for Elliptic Optimal Control Problems

H.-F. Fu, H.-X. Rui, H. Guo, J.-S. Zhang & J. Hou

Int. J. Numer. Anal. Mod., 13 (2016), pp. 610-626.

Published online: 2016-07

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  • Abstract

In this paper, we propose a splitting least-squares mixed finite element method for the approximation of elliptic optimal control problem with the control constrained by pointwise inequality. By selecting a properly least-squares minimization functional, we derive equivalent two independent, symmetric and positive definite weak formulation for the primal state variable and its flux. Then, using the first order necessary and also sufficient optimality condition, we deduce another two corresponding adjoint state equations, which are both independent, symmetric and positive definite. Also, a variational inequality for the control variable is involved. For the discretization of the state and adjoint state equations, either RT mixed finite element or standard $C^0$ finite element can be used, which is not necessary subject to the Ladyzhenkaya-Babuska-Brezzi condition. Optimal a priori error estimates in corresponding norms are derived for the control, the states and adjoint states, respectively. Finally, we use some numerical examples to validate the theoretical analysis.

  • Keywords

Optimal control, splitting least-squares, mixed finite element method, positive definite, a priori error estimates.

  • AMS Subject Headings

49K20, 49M25, 65N15, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-13-610, author = {}, title = {A Splitting Least-Squares Mixed Finite Element Method for Elliptic Optimal Control Problems}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2016}, volume = {13}, number = {4}, pages = {610--626}, abstract = {

In this paper, we propose a splitting least-squares mixed finite element method for the approximation of elliptic optimal control problem with the control constrained by pointwise inequality. By selecting a properly least-squares minimization functional, we derive equivalent two independent, symmetric and positive definite weak formulation for the primal state variable and its flux. Then, using the first order necessary and also sufficient optimality condition, we deduce another two corresponding adjoint state equations, which are both independent, symmetric and positive definite. Also, a variational inequality for the control variable is involved. For the discretization of the state and adjoint state equations, either RT mixed finite element or standard $C^0$ finite element can be used, which is not necessary subject to the Ladyzhenkaya-Babuska-Brezzi condition. Optimal a priori error estimates in corresponding norms are derived for the control, the states and adjoint states, respectively. Finally, we use some numerical examples to validate the theoretical analysis.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/455.html} }
TY - JOUR T1 - A Splitting Least-Squares Mixed Finite Element Method for Elliptic Optimal Control Problems JO - International Journal of Numerical Analysis and Modeling VL - 4 SP - 610 EP - 626 PY - 2016 DA - 2016/07 SN - 13 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/455.html KW - Optimal control, splitting least-squares, mixed finite element method, positive definite, a priori error estimates. AB -

In this paper, we propose a splitting least-squares mixed finite element method for the approximation of elliptic optimal control problem with the control constrained by pointwise inequality. By selecting a properly least-squares minimization functional, we derive equivalent two independent, symmetric and positive definite weak formulation for the primal state variable and its flux. Then, using the first order necessary and also sufficient optimality condition, we deduce another two corresponding adjoint state equations, which are both independent, symmetric and positive definite. Also, a variational inequality for the control variable is involved. For the discretization of the state and adjoint state equations, either RT mixed finite element or standard $C^0$ finite element can be used, which is not necessary subject to the Ladyzhenkaya-Babuska-Brezzi condition. Optimal a priori error estimates in corresponding norms are derived for the control, the states and adjoint states, respectively. Finally, we use some numerical examples to validate the theoretical analysis.

H.-F. Fu, H.-X. Rui, H. Guo, J.-S. Zhang & J. Hou. (1970). A Splitting Least-Squares Mixed Finite Element Method for Elliptic Optimal Control Problems. International Journal of Numerical Analysis and Modeling. 13 (4). 610-626. doi:
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