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We study an optimal control problem governed by elliptic PDEs with interface,which the control acts on the interface. Due to the jump of the coefficientacross the interface and the control acting on the interface, the regularity ofsolution of the control problem is limited on the whole domain, but smoother onsubdomains. The control function with pointwise inequality constraints isserved as the flux jump condition which we called Neumann interface control. Weuse a simple uniform mesh that is independent of the interface. The standardlinear finite element method can not achieve optimal convergence when theuniform mesh is used. Therefore the state and adjoint state equations arediscretized by piecewise linear immersed finite element method (IFEM). Whilethe accuracy of the piecewise constant approximation of the optimal control onthe interface is improved by a postprocessing step which possessessuperconvergence properties; as well as the variational discretization conceptfor the optimal control is used to improve the error estimates. Optimal errorestimates for the control, suboptimal error estimates for state and adjointstate are derived. Numerical examples with and without constraints are providedto illustrate the effectiveness of the proposed scheme and correctness of thetheoretical analysis.

A Neumann interface optimal control problem with elliptic PDE constraints and its discretization and numerical analysis