Error Estimates for Finite Element Approximation to Elliptic Optimal Control Problems with Boundary Observations in $H^{-\frac{1}{2}}(Γ)$ (Advances in Applied Mathematics and Mechanics, 2024)
In this paper we consider the finite element approximation to an ellipticoptimal control problem with boundary observations in $H^{-\frac{1}{2}}(Γ).$ This problem is motivated by certain applications in optimal control of semiconductor devices where thediscrepancy of the current density on the boundary to the target one is the objective tobe minimized. The observation in $H^{-\frac{1}{2}}(Γ)$ is realized through a Neumann-to-Dirichletmapping 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 andcontinuous finite elements. Second order convergence rate for the optimal control istheoretically proved and verified by numerical results.