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

2025/06/20 by García-Archilla, Bosco, John, V., Novo, Julia · 3 citations
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.2506.16917

openalex publication_date 2025/06/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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≤ 5, for which error bounds for q≥ 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 (Δt)^q in time for the BDF-q scheme and order h^k+1 for the L²(Ω) 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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• Viscosity in error upper bound for a consistent splitting scheme of the Navier-Stokes equations
• Stability and error analysis of IMEX-BDFk finite element schemes for the incompressible Navier-Stokes system
• A spectral-vanishing-viscosity stabilization of a higher-order consistent splitting scheme for the Navier-Stokes equations

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