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Volume 6, Issue 3
Superconvergence and $L^∞$-Error Estimates of the Lowest Order Mixed Methods for Distributed Optimal Control Problems Governed by Semilinear Elliptic Equations

Tianliang Hou

DOI: 10.4208/nmtma.2013.1133nm

Numer. Math. Theor. Meth. Appl., 6 (2013), pp. 479-498.

Published online: 2013-06

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  • Abstract

In this paper, we investigate the superconvergence property and the $L^∞$-error estimates of mixed finite element methods for a semilinear elliptic control problem. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. We derive some superconvergence results for the control variable. Moreover, we derive $L^∞$-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, distributed optimal control problems, superconvergence, $L^∞$-error estimates, mixed finite element methods.

  • AMS Subject Headings

49J20, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-6-479, author = {Tianliang Hou}, title = {Superconvergence and $L^∞$-Error Estimates of the Lowest Order Mixed Methods for Distributed Optimal Control Problems Governed by Semilinear Elliptic Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2013}, volume = {6}, number = {3}, pages = {479--498}, abstract = {

In this paper, we investigate the superconvergence property and the $L^∞$-error estimates of mixed finite element methods for a semilinear elliptic control problem. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. We derive some superconvergence results for the control variable. Moreover, we derive $L^∞$-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.2013.1133nm}, url = {http://global-sci.org/intro/article_detail/nmtma/5914.html} }
TY - JOUR T1 - Superconvergence and $L^∞$-Error Estimates of the Lowest Order Mixed Methods for Distributed Optimal Control Problems Governed by Semilinear Elliptic Equations AU - Tianliang Hou JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 479 EP - 498 PY - 2013 DA - 2013/06 SN - 6 DO - http://doi.org/10.4208/nmtma.2013.1133nm UR - https://global-sci.org/intro/article_detail/nmtma/5914.html KW - Semilinear elliptic equations, distributed optimal control problems, superconvergence, $L^∞$-error estimates, mixed finite element methods. AB -

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

Tianliang Hou. (2013). Superconvergence and $L^∞$-Error Estimates of the Lowest Order Mixed Methods for Distributed Optimal Control Problems Governed by Semilinear Elliptic Equations. Numerical Mathematics: Theory, Methods and Applications. 6 (3). 479-498. doi:10.4208/nmtma.2013.1133nm
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