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Volume 16, Issue 5
Error Estimates for Finite Element Approximation to Elliptic Optimal Control Problems with Boundary Observations in $H^{-\frac{1}{2}}(\Gamma)$

Xuelin Tao

Adv. Appl. Math. Mech., 16 (2024), pp. 1152-1175.

Published online: 2024-07

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  • Abstract

In this paper we consider the finite element approximation to an elliptic optimal control problem with boundary observations in $H^{-\frac{1}{2}}(\Gamma).$ This problem is motivated by certain applications in optimal control of semiconductor devices where the discrepancy of the current density on the boundary to the target one is the objective to be minimized. The observation in $H^{-\frac{1}{2}}(\Gamma)$ is realized through a Neumann-to-Dirichlet mapping which facilitates the theoretical and numerical analysis. A priori error estimate for the optimal control is derived based on the variational control discretization, whereas the state and adjoint state variables are approximated by piecewise linear and continuous finite elements. Second order convergence rate for the optimal control is theoretically proved and verified by numerical results.

  • AMS Subject Headings

49M25, 49M41, 49K20, 65N30

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COPYRIGHT: © Global Science Press

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@Article{AAMM-16-1152, author = {Tao , Xuelin}, title = {Error Estimates for Finite Element Approximation to Elliptic Optimal Control Problems with Boundary Observations in $H^{-\frac{1}{2}}(\Gamma)$}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2024}, volume = {16}, number = {5}, pages = {1152--1175}, abstract = {

In this paper we consider the finite element approximation to an elliptic optimal control problem with boundary observations in $H^{-\frac{1}{2}}(\Gamma).$ This problem is motivated by certain applications in optimal control of semiconductor devices where the discrepancy of the current density on the boundary to the target one is the objective to be minimized. The observation in $H^{-\frac{1}{2}}(\Gamma)$ is realized through a Neumann-to-Dirichlet mapping which facilitates the theoretical and numerical analysis. A priori error estimate for the optimal control is derived based on the variational control discretization, whereas the state and adjoint state variables are approximated by piecewise linear and continuous finite elements. Second order convergence rate for the optimal control is theoretically proved and verified by numerical results.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2022-0172}, url = {http://global-sci.org/intro/article_detail/aamm/23289.html} }
TY - JOUR T1 - Error Estimates for Finite Element Approximation to Elliptic Optimal Control Problems with Boundary Observations in $H^{-\frac{1}{2}}(\Gamma)$ AU - Tao , Xuelin JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 1152 EP - 1175 PY - 2024 DA - 2024/07 SN - 16 DO - http://doi.org/10.4208/aamm.OA-2022-0172 UR - https://global-sci.org/intro/article_detail/aamm/23289.html KW - Optimal control problems, boundary observations, finite element methods, error estimate. AB -

In this paper we consider the finite element approximation to an elliptic optimal control problem with boundary observations in $H^{-\frac{1}{2}}(\Gamma).$ This problem is motivated by certain applications in optimal control of semiconductor devices where the discrepancy of the current density on the boundary to the target one is the objective to be minimized. The observation in $H^{-\frac{1}{2}}(\Gamma)$ is realized through a Neumann-to-Dirichlet mapping which facilitates the theoretical and numerical analysis. A priori error estimate for the optimal control is derived based on the variational control discretization, whereas the state and adjoint state variables are approximated by piecewise linear and continuous finite elements. Second order convergence rate for the optimal control is theoretically proved and verified by numerical results.

Xuelin Tao. (2024). Error Estimates for Finite Element Approximation to Elliptic Optimal Control Problems with Boundary Observations in $H^{-\frac{1}{2}}(\Gamma)$. Advances in Applied Mathematics and Mechanics. 16 (5). 1152-1175. doi:10.4208/aamm.OA-2022-0172
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