Error Analysis of BDF 1–6 Time-Stepping Methods for the Transient Stokes Problem: Velocity and Pressure Estimates

SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1586-1616, August 2025.
Abstract. We present a new stability and error analysis of fully discrete approximation schemes for the transient Stokes equation. For the spatial discretization, we consider a wide class of Galerkin finite element methods which includes both inf-sup stable spaces and symmetric pressure stabilized formulations. We extend the results from Burman and Fernández [SIAM J. Numer. Anal., 47 (2009), pp. 409–439] and provide a unified theoretical analysis of backward difference formula methods of orders 1 to 6. The main novelty of our approach lies in deriving optimal-order stability and error estimates for both the velocity and the pressure using Dahlquist’s [math]-stability concept together with the multiplier technique introduced by Nevanlinna and Odeh and recently by Akrivis et al. [SIAM J. Numer. Anal., 59 (2021), pp. 2449–2472]. When combined with a method-dependent Ritz projection for the initial data, unconditional stability can be shown, while for arbitrary interpolation, pressure stability is subordinate to the fulfillment of a mild inverse CFL-type condition between space and time discretizations.