Convergence analysis of variable-time-step BDF2/spectral approximations for optimal control problems governed by linear reaction-diffusion equations

Abstract

In this paper, we focus on the optimal control problem governed by a linear reaction-diffusion equation with constraints on the control variable. We construct an effective fully-discrete scheme to solve this problem by using the variable-time-step two-step backward differentiation formula (VSBDF2) in time combining with the Galerkin spectral methods in space. By using the recently developed techniques including the discrete orthogonal convolution (DOC) kernels, and the positive definiteness of BDF2 convolution kernels, we obtain an error estimate of order $O({\tau }^2+N^{-p})$ under a mild restriction $\frac{1}{4.8645}\le r_k\le 4.8645$ for the ratio of adjacent time steps $r_k$, where $\tau ,N,p$ are the maximum time step size, polynomial degree, and regularity of the exact solution respectively. Several numerical examples are provided to validate the theoretical results and to demonstrate the efficiency of the proposed method.