Immersed finite element method and its analysis for parabolic optimal control problems with interfaces

Abstract

In this paper, we develop the immersed finite element method for parabolic optimal control problems with interfaces. By employing the definition of directional derivative of Lagrange function, first-order necessary optimality conditions in qualified form for parabolic optimal control problems with interfaces are established. The parabolic state equations are treated with the immersed finite element method using a simple uniform mesh which is independent of the interface. In the interface elements, functions are locally piecewise bilinear functions according to the subelements formed by the actual interface curves. By introducing the auxiliary functions which are the solutions of interface parabolic equations with non-homogeneous and homogeneous jump conditions, optimal error estimates are proved for the proposed schemes to the controls, states and adjoint states in both the semi-discrete case and the fully discrete case. Numerical experiments show the performance of the proposed scheme to solve the parabolic optimal control problems with interfaces and confirm the theoretical results.