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.

翻译：本文针对瞬态Stokes方程，提出了一种全离散逼近格式的稳定性和误差新分析。在空间离散方面，我们考虑了包含inf-sup稳定空间和对称压力稳定公式在内的广泛伽辽金有限元方法。我们将Burman和Fernández [《SIAM J. Numer. Anal.》，第47卷，2009年，第409-439页]的结果进行推广，为阶数1至6的向后差分公式（BDF方法）提供了统一的理论分析。该方法的主要创新在于，结合Dahlquist的G稳定性概念与Nevannlina-Odeh及最近Akrivis等人[《SIAM J. Numer. Anal.》，第59卷，2021年，第2449-2472页]引入的乘子技巧，推导出速度和压力的最优稳定性及误差估计。当与初始数据的Ritz投影方法相结合时，可证明无条件稳定性；而对于任意插值，压力稳定性取决于空间与时间离散之间满足一个温和的逆CFL型条件。