The Adaptive Immersed Finite Element Method for the Variational Discretization of the Elliptic Optimal Control Problem with Interface

Abstract

In this paper, we develop the adaptive immersed interface finite element method based on the variational discretization for solving optimal control problems governed by elliptic PDEs with interface. The meshes in this method do not need to fit the interface. The state and co-state variables are discretized by piecewise linear continuous functions and the control variable is required in the variational discretization approach. A posteriori error estimates for the control, state and adjoint state are derived. New error indicators are introduced to control the error due to non-body-fitted meshes. The error estimators are implemented and tested with promising numerical results which will show the competitive behavior of the adaptive algorithm.