Error analysis of BDF 1-6 time-stepping methods for the transient Stokes problem: velocity and pressure estimates

TL;DR

This paper provides a comprehensive stability and error analysis of BDF time-stepping methods (orders 1-6) for the transient Stokes problem, including velocity and pressure estimates, using advanced theoretical techniques.

Contribution

It extends previous analyses by applying Dahlquist's G-stability and multiplier techniques to derive unified optimal stability and error estimates for BDF methods in the transient Stokes problem.

Findings

Unconditional stability with Ritz projection for initial data.

Pressure stability under mild inverse CFL condition.

Optimal error estimates for velocity and pressure.

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 Fern\'andez [\textit{SIAM J. Numer. Anal.}, 47 (2009), pp. 409-439] and provide a unified theoretical analysis of backward difference formulae (BDF methods) of order 1 to 6. The main novelty of our approach lies in the use of Dahlquist's G-stability concept together with multiplier techniques introduced by Nevannlina-Odeh and recently by Akrivis et al. [\textit{SIAM J. Numer. Anal.}, 59 (2021), pp. 2449-2472] to derive optimal stability and error estimates for both the velocity and the pressure. 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.