A priori error estimates of VSBDF2 schemes for solving parabolic distributed optimal control problems

In this article, the focus is on utilizing the variable-step BDF2 (VSBDF2) schemes in combination with the finite element method (FEM) to solve parabolic distributed optimal control problems. First of all, with the help of the Newton backward and forward interpolating polynomials, we propose the VSBDF2 backward and forward formulas, respectively. Then, similar to the discrete orthogonal convolution (DOC) backward kernels, we define the novel discrete orthogonal convolution (DOC) forward kernels here. Via using a new theoretical framework with DOC backward and forward kernels, the VSBDF2 backward and forward formulas are reconsidered, respectively. We establish that if the ratios between adjacent time-steps are bounded by 2/(3+17)⩽rk≔τk/τk−1⩽(3+17)/2, the VSBDF2 schemes will show at least first-order temporal convergence. Furthermore, if nearly all of ratios are within the range 1/(1+2)⩽rk⩽1+2 or certain high-order initial schemes are employed, we can derive a priori error estimates with second-order temporal accuracy for parabolic distributed optimal control problems. Finally, we present some numerical examples aimed at verifying the theoretical findings. •Propose the VSBDF2 backward and forward formulas.•Define a novel DOC forward kernels and develop some new properties.•Reconsider the VSBDF2 formulas via using the DOC kernels.•Utilize the VSBDF2 formulas with FEM to solve parabolic optimal control problems.•Derive a priori error estimates with second-order temporal accuracy.