Crank–Nicolson finite difference schemes for parabolic optimal Dirichlet boundary control problems

Abstract

In this paper, we adopt the optimize‐then‐discretize approach to solve parabolic optimal Dirichlet boundary control problems. First, we derive the first‐order necessary optimality system, which includes the state, co‐state equations, and the optimality condition. Then, we propose Crank–Nicolson finite difference schemes to discretize the optimality system in 1D and 2D cases, respectively. In order to build the second‐order spatial approximation, we use the ghost points on the boundary in the schemes. We prove that the proposed schemes are unconditionally stable, compatible, and second‐order convergent in both time and space. To avoid solving the large coupled schemes directly, we use the iterative method. Finally, we present several numerical examples to validate our theoretical analysis.