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Volume 6, Issue 4
Error Estimates and Superconvergence of RT0 Mixed Methods for a Class of Semilinear Elliptic Optimal Control Problems

Yanping Chen & Tianliang Hou

DOI: 10.4208/nmtma.2013.1230nm

Numer. Math. Theor. Meth. Appl., 6 (2013), pp. 637-656.

Published online: 2013-06

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  • Abstract

In this paper, we will investigate the error estimates and the superconvergence property of mixed finite element methods for a semilinear elliptic control problem with an integral constraint on control. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element and the control variable is approximated by piecewise constant functions. We derive some superconvergence properties for the control variable and the state variables. Moreover, we derive $L^∞$- and $H^{-1}$-error estimates both for the control variable and the state variables. Finally, a numerical example is given to demonstrate the theoretical results.

  • Keywords

Semilinear elliptic equations, optimal control problems, superconvergence, error estimates, mixed finite element methods.

  • AMS Subject Headings

49J20, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-6-637, author = {Yanping Chen and Tianliang Hou}, title = {Error Estimates and Superconvergence of RT0 Mixed Methods for a Class of Semilinear Elliptic Optimal Control Problems}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2013}, volume = {6}, number = {4}, pages = {637--656}, abstract = {

In this paper, we will investigate the error estimates and the superconvergence property of mixed finite element methods for a semilinear elliptic control problem with an integral constraint on control. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element and the control variable is approximated by piecewise constant functions. We derive some superconvergence properties for the control variable and the state variables. Moreover, we derive $L^∞$- and $H^{-1}$-error estimates both for the control variable and the state variables. Finally, a numerical example is given to demonstrate the theoretical results.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2013.1230nm}, url = {http://global-sci.org/intro/article_detail/nmtma/5923.html} }
TY - JOUR T1 - Error Estimates and Superconvergence of RT0 Mixed Methods for a Class of Semilinear Elliptic Optimal Control Problems AU - Yanping Chen & Tianliang Hou JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 637 EP - 656 PY - 2013 DA - 2013/06 SN - 6 DO - http://doi.org/10.4208/nmtma.2013.1230nm UR - https://global-sci.org/intro/article_detail/nmtma/5923.html KW - Semilinear elliptic equations, optimal control problems, superconvergence, error estimates, mixed finite element methods. AB -

In this paper, we will investigate the error estimates and the superconvergence property of mixed finite element methods for a semilinear elliptic control problem with an integral constraint on control. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element and the control variable is approximated by piecewise constant functions. We derive some superconvergence properties for the control variable and the state variables. Moreover, we derive $L^∞$- and $H^{-1}$-error estimates both for the control variable and the state variables. Finally, a numerical example is given to demonstrate the theoretical results.

Yanping Chen and Tianliang Hou. (2013). Error Estimates and Superconvergence of RT0 Mixed Methods for a Class of Semilinear Elliptic Optimal Control Problems. Numerical Mathematics: Theory, Methods and Applications. 6 (4). 637-656. doi:10.4208/nmtma.2013.1230nm
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