Error estimates of Fourier finite volume element method for parabolic Dirichlet boundary optimal control problems on complex connected domains

Abstract

In this paper, we study the parabolic Dirichlet boundary optimal control on complex connected domains. It is well known that both the complex connected domain and the Dirichlet boundary control bring great difficulties to theoretical analysis and numerical calculation. The complex connected domain is a typical non-convex domain, and it is difficult for the traditional numerical method to obtain the same convergence order as the state and adjoint state for the Dirichlet boundary control. The optimal system of the proposed control problem is first obtained by using Lagrange multiplier method. Then, based on the variational form, the Fourier finite volume element method is used to obtain the fully discrete scheme of the optimality system, that is, using Fourier expansion method in azimuthal direction and applying finite volume element method in radial direction respectively, and the Crank-Nicolson scheme in the time direction. Next, we strictly prove the error estimates of state, adjoint state and Dirichlet boundary control by using the variational discretization concept. Finally, the theoretical analysis results and the feasibility and effectiveness of the proposed method are verified by numerical experiments.