Immersed finite elements for optimal control problems of elliptic PDEs with interfaces

This paper presents a numerical method and analysis, based on the variational discretization concept, for optimal control problems governed by elliptic PDEs with interfaces. The method uses a simple uniform mesh which is independent of the interface. Due to the jump of the coefficient across the interface, the standard linear finite element method cannot achieve optimal convergence when the uniform mesh is used. Therefore the immersed finite element method (IFEM) developed in Li et al. [20] is used to discretize the state equation required in the variational discretization approach. Optimal error estimates for the control, state and adjoint state are derived. Numerical examples are provided to confirm the theoretical results.