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

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 Benard 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 Benard problem Nonlinear coupled system Finite element approximation Superconvergence

49J20 65N30.

@Article{JCM-26-660, author = {}, title = {Superconvergence Analysis of Finite Element Methods for Optimal Control Problems of the Stationary Benard 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 Benard 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 Benard Type 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 KW - The stationary Benard problem KW - Nonlinear coupled system KW - Finite element approximation KW - Superconvergence AB - In this paper, we consider the finite element approximation of the distributed optimal control problems of the stationary Benard 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.