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Volume 42, Issue 6
Error Analysis for Parabolic Optimal Control Problems with Measure Data in a Nonconvex Polygonal Domain

Pratibha Shakya

J. Comp. Math., 42 (2024), pp. 1579-1604.

Published online: 2024-11

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  • Abstract

This paper considers the finite element approximation to parabolic optimal control problems with measure data in a nonconvex polygonal domain. Such problems usually possess low regularity in the state variable due to the presence of measure data and the nonconvex nature of the domain. The low regularity of the solution allows the finite element approximations to converge at lower orders. We prove the existence, uniqueness and regularity results for the solution to the control problem satisfying the first order optimality condition. For our error analysis we have used piecewise linear elements for the approximation of the state and co-state variables, whereas piecewise constant functions are employed to approximate the control variable. The temporal discretization is based on the implicit Euler scheme. We derive both a priori and a posteriori error bounds for the state, control and co-state variables. Numerical experiments are performed to validate the theoretical rates of convergence.

  • AMS Subject Headings

49J20, 49K20, 65N15, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-42-1579, author = {Shakya , Pratibha}, title = {Error Analysis for Parabolic Optimal Control Problems with Measure Data in a Nonconvex Polygonal Domain}, journal = {Journal of Computational Mathematics}, year = {2024}, volume = {42}, number = {6}, pages = {1579--1604}, abstract = {

This paper considers the finite element approximation to parabolic optimal control problems with measure data in a nonconvex polygonal domain. Such problems usually possess low regularity in the state variable due to the presence of measure data and the nonconvex nature of the domain. The low regularity of the solution allows the finite element approximations to converge at lower orders. We prove the existence, uniqueness and regularity results for the solution to the control problem satisfying the first order optimality condition. For our error analysis we have used piecewise linear elements for the approximation of the state and co-state variables, whereas piecewise constant functions are employed to approximate the control variable. The temporal discretization is based on the implicit Euler scheme. We derive both a priori and a posteriori error bounds for the state, control and co-state variables. Numerical experiments are performed to validate the theoretical rates of convergence.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2305-m2022-0215}, url = {http://global-sci.org/intro/article_detail/jcm/23508.html} }
TY - JOUR T1 - Error Analysis for Parabolic Optimal Control Problems with Measure Data in a Nonconvex Polygonal Domain AU - Shakya , Pratibha JO - Journal of Computational Mathematics VL - 6 SP - 1579 EP - 1604 PY - 2024 DA - 2024/11 SN - 42 DO - http://doi.org/10.4208/jcm.2305-m2022-0215 UR - https://global-sci.org/intro/article_detail/jcm/23508.html KW - A priori and a posteriori error estimates, Finite element method, Measure data, Nonconvex polygonal domain, Optimal control problem. AB -

This paper considers the finite element approximation to parabolic optimal control problems with measure data in a nonconvex polygonal domain. Such problems usually possess low regularity in the state variable due to the presence of measure data and the nonconvex nature of the domain. The low regularity of the solution allows the finite element approximations to converge at lower orders. We prove the existence, uniqueness and regularity results for the solution to the control problem satisfying the first order optimality condition. For our error analysis we have used piecewise linear elements for the approximation of the state and co-state variables, whereas piecewise constant functions are employed to approximate the control variable. The temporal discretization is based on the implicit Euler scheme. We derive both a priori and a posteriori error bounds for the state, control and co-state variables. Numerical experiments are performed to validate the theoretical rates of convergence.

Shakya , Pratibha. (2024). Error Analysis for Parabolic Optimal Control Problems with Measure Data in a Nonconvex Polygonal Domain. Journal of Computational Mathematics. 42 (6). 1579-1604. doi:10.4208/jcm.2305-m2022-0215
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