Volume 4, Issue 6
Error Estimates and Superconvergence of Mixed Finite Element Methods for Optimal Control Problems with Low Regularity

Adv. Appl. Math. Mech., 4 (2012), pp. 751-768.

Published online: 2012-12

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• Abstract

In this paper, we investigate the error estimates and superconvergence property of mixed finite element methods for elliptic optimal control problems. 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 $L^2$ and $L^\infty$-error estimates for the control variable. Moreover, using a recovery operator, we also derive some superconvergence results for the control variable. Finally, a numerical example is given to demonstrate the theoretical results.

• Keywords

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

49J20, 65N30

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@Article{AAMM-4-751, author = {Yanping Chen , and Tianliang Hou , and Zheng , Weishan}, title = {Error Estimates and Superconvergence of Mixed Finite Element Methods for Optimal Control Problems with Low Regularity}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2012}, volume = {4}, number = {6}, pages = {751--768}, abstract = {

In this paper, we investigate the error estimates and superconvergence property of mixed finite element methods for elliptic optimal control problems. 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 $L^2$ and $L^\infty$-error estimates for the control variable. Moreover, using a recovery operator, we also derive some superconvergence results for the control variable. Finally, a numerical example is given to demonstrate the theoretical results.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.12-12S05}, url = {http://global-sci.org/intro/article_detail/aamm/147.html} }
TY - JOUR T1 - Error Estimates and Superconvergence of Mixed Finite Element Methods for Optimal Control Problems with Low Regularity AU - Yanping Chen , AU - Tianliang Hou , AU - Zheng , Weishan JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 751 EP - 768 PY - 2012 DA - 2012/12 SN - 4 DO - http://doi.org/10.4208/aamm.12-12S05 UR - https://global-sci.org/intro/article_detail/aamm/147.html KW - Elliptic equations, optimal control problems, superconvergence, error estimates, mixed finite element methods. AB -

In this paper, we investigate the error estimates and superconvergence property of mixed finite element methods for elliptic optimal control problems. 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 $L^2$ and $L^\infty$-error estimates for the control variable. Moreover, using a recovery operator, we also derive some superconvergence results for the control variable. Finally, a numerical example is given to demonstrate the theoretical results.

Yanping Chen, Tianliang Hou & Weishan Zheng. (1970). Error Estimates and Superconvergence of Mixed Finite Element Methods for Optimal Control Problems with Low Regularity. Advances in Applied Mathematics and Mechanics. 4 (6). 751-768. doi:10.4208/aamm.12-12S05
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