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Error analysis of BDF schemes for the evolutionary incompressible Navier--Stokes equations

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arxiv 2506.16917 v1 pith:QRS4Z3O3 submitted 2025-06-20 math.NA cs.NA

Bosco García-Archilla , V. John , Julia Novo This is my paper · ORCID
classification math.NAcs.NA
keywords errorboundsvelocitybdf-caseequationsevolutionaryfinite
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abstract

Error bounds for fully discrete schemes for the evolutionary incompressible Navier--Stokes equations are derived in this paper. For the time integration we apply BDF-$q$ methods, $q\le 5$, for which error bounds for $q\ge 3$ cannot be found in the literature. Inf-sup stable mixed finite elements are used as spatial approximation. First, we analyze the standard Galerkin method and second a grad-div stabilized method. The grad-div stabilization allows to prove error bounds with constants independent of inverse powers of the viscosity coefficient. We prove optimal bounds for the velocity and pressure with order $(\Delta t)^q$ in time for the BDF-$q$ scheme and order $h^{k+1}$ for the $L^2(\Omega)$ error of the velocity in the first case and $h^k$ in the second case, $k$ being the degree of the polynomials in finite element velocity space.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Unconditionally Long-Time Stable Variable-Step Second-Order ETD Schemes for the 2D Periodic Incompressible NSE

    math.NA 2026-02 conditional novelty 7.0 of 10

    A variable-step second-order ETD-mr-SAV scheme for the periodic NSE is claimed to be uniformly long-time stable, but the proof's key sum bound is erroneous and second-order accuracy is only argued heuristically.

  2. A spectral-vanishing-viscosity stabilization of a higher-order consistent splitting scheme for the Navier-Stokes equations

    math.NA 2026-07 accept novelty 6.0 of 10

    Directional Maday–Kaber–Tadmor SVV stabilizes Huang–Shen high-order BDF–IMEX consistent splitting for Navier–Stokes at high Re, with stability/error estimates and 2D validation.

  3. Second order reduced model via incremental projection for Navier Stokes

    math.NA 2025-12 reject novelty 4.0 of 10

    A POD-based reduced-order model for the unsteady Stokes equations using a BDF2 incremental projection scheme is claimed to be second-order accurate, but the printed numerical validation and reduced system contain errors.

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