A posteriori error estimates for higher-order BDF methods for parabolic partial differential equations

We derive a posteriori error estimates for parabolic problems discretized by conforming finite elements in space and by the backward differentiation formula of order k in time, with variable step-sizes for k ≤ 3 and constant step-size for $k = 4$. The key ingredient of the analysis is a polynomial $(k+1)$-point time reconstruction of degree k of the approximate solution combined with the elliptic reconstruction. We conclude upper a posteriori error bounds in L^∞(L²) ∩ L²(H¹) and in L^∞(H¹) ∩ H¹(L²). Numerical experiments for k = 1, , 4 with linear and quadratic finite elements confirm the optimal orders of the estimators in time and in space.