Error Estimates for Finite Element Approximation to Elliptic Optimal Control Problems with Boundary Observations in $H^{-\frac{1}{2}}(\Gamma)$
Year: 2024
Author: Xuelin Tao
Advances in Applied Mathematics and Mechanics, Vol. 16 (2024), Iss. 5 : pp. 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/aamm.OA-2022-0172
Advances in Applied Mathematics and Mechanics, Vol. 16 (2024), Iss. 5 : pp. 1152–1175
Published online: 2024-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 24
Keywords: Optimal control problems boundary observations finite element methods error estimate.