Robust second-order VSBDF2 finite element schemes for parabolic distributed optimal control problems

In this paper, we further investigate the variable-step BDF2 (VSBDF2) finite element schemes for solving control constrained distributed optimal control problems governed by parabolic equations, where the state, co-state and control variables are approximated by piecewise linear functions. We first propose the VSBDF2 backward and forward formulas, respectively. Then, motivated by the discrete orthogonal convolution (DOC) and discrete complementary convolution (DCC) backward kernels (Liao and Zhang, 2021; Zhang and Zhao, 2022), we introduce the novel DOC and DCC forward kernels. Utilizing the new analytical tool of backward and forward kernels concerning the DOC and DCC, we can obtain a priori error estimates with the optimal second-order temporal accuracy for the parabolic distributed optimal control problem under the restriction condition $1/4.8645⩽r_k⩽4.8645$. Moreover, our analysis also shows that the initial solutions $y^1$ and $p^{N-1}$ obtained by VSBDF1 backward and forward formulas (i.e., the variable-step backward and forward Euler formulas) do not result in the loss of accuracy with $1/4.8645⩽r_k⩽4.8645$. Numerical experiments are provided to validate our theoretical analysis.