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arXiv · 2312.05511

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

Alessandro Contri·Balázs Kovács·André Massing
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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 [\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.

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Alessandro Contri, Balázs Kovács, André Massing. 2023-12-09. Error analysis of BDF 1-6 time-stepping methods for the transient Stokes problem: velocity and pressure estimates. https://arxiv.org/abs/2312.05511

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