Optimal a posteriori estimators for the variable step-size BDF2 method for linear parabolic equations

Optimal a posteriori error estimates for time discretizations of linear parabolic equations by the two-step backward differentiation formula (BDF2) method with variable step-sizes are derived. Based on second-order BDF reconstructions of the piecewise linear approximate solutions, the optimality of residual-based a posteriori error estimators is proved by using a novel stability inequality when the starting value is computed by the trapezoidal method. With a reasonable choice for the starting step-size, the optimality of the estimators when the starting value is computed by the backward Euler scheme can be also ensured. The effectiveness of the a posteriori error estimators is illustrated by a numerical example.