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Volume 13, Issue 4
On Discontinuous Finite Volume Approximations for Semilinear Parabolic Optimal Control Problems

R. Sandilya & S. Kumar

Int. J. Numer. Anal. Mod., 13 (2016), pp. 545-568.

Published online: 2016-07

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  • Abstract

In this article, we discuss and analyze discontinuous finite volume approximations of the distributed optimal control problems governed by a class of semilinear parabolic partial differential equations with control constraints. For the spatial discretization of the state and costate variables, piecewise linear elements are used and an implicit finite difference scheme is used for time derivatives; whereas, for the approximation of the control variable, three different strategies are used: variational discretization, piecewise constant and piecewise linear discretization. A priori error estimates (for these three approaches) in suitable $L^2$-norm are derived for state, co-state and control variables. Numerical experiments are presented in order to assure the accuracy and rate of the convergence of the proposed scheme.

  • Keywords

Semilinear parabolic optimal control problems, variational discretization, piecewise constant and piecewise linear discretization, discontinuous finite volume methods, a priori error estimates, numerical experiments.

  • AMS Subject Headings

65Q05

  • Copyright

COPYRIGHT: © Global Science Press

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  • TXT
@Article{IJNAM-13-545, author = {R. Sandilya and S. Kumar}, title = {On Discontinuous Finite Volume Approximations for Semilinear Parabolic Optimal Control Problems}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2016}, volume = {13}, number = {4}, pages = {545--568}, abstract = {

In this article, we discuss and analyze discontinuous finite volume approximations of the distributed optimal control problems governed by a class of semilinear parabolic partial differential equations with control constraints. For the spatial discretization of the state and costate variables, piecewise linear elements are used and an implicit finite difference scheme is used for time derivatives; whereas, for the approximation of the control variable, three different strategies are used: variational discretization, piecewise constant and piecewise linear discretization. A priori error estimates (for these three approaches) in suitable $L^2$-norm are derived for state, co-state and control variables. Numerical experiments are presented in order to assure the accuracy and rate of the convergence of the proposed scheme.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/452.html} }
TY - JOUR T1 - On Discontinuous Finite Volume Approximations for Semilinear Parabolic Optimal Control Problems AU - R. Sandilya & S. Kumar JO - International Journal of Numerical Analysis and Modeling VL - 4 SP - 545 EP - 568 PY - 2016 DA - 2016/07 SN - 13 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/452.html KW - Semilinear parabolic optimal control problems, variational discretization, piecewise constant and piecewise linear discretization, discontinuous finite volume methods, a priori error estimates, numerical experiments. AB -

In this article, we discuss and analyze discontinuous finite volume approximations of the distributed optimal control problems governed by a class of semilinear parabolic partial differential equations with control constraints. For the spatial discretization of the state and costate variables, piecewise linear elements are used and an implicit finite difference scheme is used for time derivatives; whereas, for the approximation of the control variable, three different strategies are used: variational discretization, piecewise constant and piecewise linear discretization. A priori error estimates (for these three approaches) in suitable $L^2$-norm are derived for state, co-state and control variables. Numerical experiments are presented in order to assure the accuracy and rate of the convergence of the proposed scheme.

R. Sandilya and S. Kumar. (2016). On Discontinuous Finite Volume Approximations for Semilinear Parabolic Optimal Control Problems. International Journal of Numerical Analysis and Modeling. 13 (4). 545-568. doi:
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