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Volume 5, Issue 3
Superconvergence and $L^{\infty}$-Error Estimates of RT1 Mixed Methods for Semilinear Elliptic Control Problems with an Integral Constraint

Yanping Chen & Tianliang Hou

DOI: 10.4208/nmtma.2012.m1118

Numer. Math. Theor. Meth. Appl., 5 (2012), pp. 423-446.

Published online: 2012-05

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  • Abstract

In this paper, we investigate the superconvergence property and the $L^{\infty}$-error estimates of mixed finite element methods for a semilinear elliptic control problem with an integral constraint. The state and co-state are approximated by the order one Raviart-Thomas mixed finite element space and the control variable is approximated by piecewise constant functions or piecewise linear functions. We derive some superconvergence results for the control variable and the state variables when the control is approximated by piecewise constant functions. Moreover, we derive $L^{\infty}$-error estimates for both the control variable and the state variables when the control is discretized by piecewise linear functions. Finally, some numerical examples are given to demonstrate the theoretical results.  

  • Keywords

Semilinear elliptic equations, optimal control problems, superconvergence, $L^{\infty}$-error estimates, mixed finite element methods, postprocessing.

  • AMS Subject Headings

49J20, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-5-423, author = {Yanping Chen and Tianliang Hou}, title = {Superconvergence and $L^{\infty}$-Error Estimates of RT1 Mixed Methods for Semilinear Elliptic Control Problems with an Integral Constraint}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2012}, volume = {5}, number = {3}, pages = {423--446}, abstract = {

In this paper, we investigate the superconvergence property and the $L^{\infty}$-error estimates of mixed finite element methods for a semilinear elliptic control problem with an integral constraint. The state and co-state are approximated by the order one Raviart-Thomas mixed finite element space and the control variable is approximated by piecewise constant functions or piecewise linear functions. We derive some superconvergence results for the control variable and the state variables when the control is approximated by piecewise constant functions. Moreover, we derive $L^{\infty}$-error estimates for both the control variable and the state variables when the control is discretized by piecewise linear functions. Finally, some numerical examples are given to demonstrate the theoretical results.  

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2012.m1118}, url = {http://global-sci.org/intro/article_detail/nmtma/5945.html} }
TY - JOUR T1 - Superconvergence and $L^{\infty}$-Error Estimates of RT1 Mixed Methods for Semilinear Elliptic Control Problems with an Integral Constraint AU - Yanping Chen & Tianliang Hou JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 423 EP - 446 PY - 2012 DA - 2012/05 SN - 5 DO - http://doi.org/10.4208/nmtma.2012.m1118 UR - https://global-sci.org/intro/article_detail/nmtma/5945.html KW - Semilinear elliptic equations, optimal control problems, superconvergence, $L^{\infty}$-error estimates, mixed finite element methods, postprocessing. AB -

In this paper, we investigate the superconvergence property and the $L^{\infty}$-error estimates of mixed finite element methods for a semilinear elliptic control problem with an integral constraint. The state and co-state are approximated by the order one Raviart-Thomas mixed finite element space and the control variable is approximated by piecewise constant functions or piecewise linear functions. We derive some superconvergence results for the control variable and the state variables when the control is approximated by piecewise constant functions. Moreover, we derive $L^{\infty}$-error estimates for both the control variable and the state variables when the control is discretized by piecewise linear functions. Finally, some numerical examples are given to demonstrate the theoretical results.  

Yanping Chen and Tianliang Hou. (2012). Superconvergence and $L^{\infty}$-Error Estimates of RT1 Mixed Methods for Semilinear Elliptic Control Problems with an Integral Constraint. Numerical Mathematics: Theory, Methods and Applications. 5 (3). 423-446. doi:10.4208/nmtma.2012.m1118
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