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Volume 26, Issue 5
Superconvergence Analysis of Finite Element Methods for Optimal Control Problems of the Stationary Bénard Type

Yanzhen Chang & Danping Yang

J. Comp. Math., 26 (2008), pp. 660-676.

Published online: 2008-10

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  • Abstract

In this paper, we consider the finite element approximation of the distributed optimal control problems of the stationary Bénard type under the pointwise control constraint. The states and the co-states are approximated by polynomial functions of lowest-order mixed finite element space or piecewise linear functions and the control is approximated by piecewise constant functions. We give the superconvergence analysis for the control; it is proved that the approximation has a second-order rate of convergence. We further give the superconvergence analysis for the states and the co-states. Then we derive error estimates in $L^\infty$-norm and optimal error estimates in $L^2$-norm.

  • Keywords

Optimal control problem, The stationary Bénard problem, Nonlinear coupled system, Finite element approximation, Superconvergence.

  • AMS Subject Headings

49J20, 65N30.

  • Copyright

COPYRIGHT: © Global Science Press

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  • TXT
@Article{JCM-26-660, author = {Yanzhen Chang and Danping Yang}, title = {Superconvergence Analysis of Finite Element Methods for Optimal Control Problems of the Stationary Bénard Type}, journal = {Journal of Computational Mathematics}, year = {2008}, volume = {26}, number = {5}, pages = {660--676}, abstract = {

In this paper, we consider the finite element approximation of the distributed optimal control problems of the stationary Bénard type under the pointwise control constraint. The states and the co-states are approximated by polynomial functions of lowest-order mixed finite element space or piecewise linear functions and the control is approximated by piecewise constant functions. We give the superconvergence analysis for the control; it is proved that the approximation has a second-order rate of convergence. We further give the superconvergence analysis for the states and the co-states. Then we derive error estimates in $L^\infty$-norm and optimal error estimates in $L^2$-norm.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8650.html} }
TY - JOUR T1 - Superconvergence Analysis of Finite Element Methods for Optimal Control Problems of the Stationary Bénard Type AU - Yanzhen Chang & Danping Yang JO - Journal of Computational Mathematics VL - 5 SP - 660 EP - 676 PY - 2008 DA - 2008/10 SN - 26 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8650.html KW - Optimal control problem, The stationary Bénard problem, Nonlinear coupled system, Finite element approximation, Superconvergence. AB -

In this paper, we consider the finite element approximation of the distributed optimal control problems of the stationary Bénard type under the pointwise control constraint. The states and the co-states are approximated by polynomial functions of lowest-order mixed finite element space or piecewise linear functions and the control is approximated by piecewise constant functions. We give the superconvergence analysis for the control; it is proved that the approximation has a second-order rate of convergence. We further give the superconvergence analysis for the states and the co-states. Then we derive error estimates in $L^\infty$-norm and optimal error estimates in $L^2$-norm.

Yanzhen Chang and Danping Yang. (2008). Superconvergence Analysis of Finite Element Methods for Optimal Control Problems of the Stationary Bénard Type. Journal of Computational Mathematics. 26 (5). 660-676. doi:
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