2025/09/15 by Franziska Weber, Weber, Franziska
Mathematics · #65M12 #65M60 #76D05 #76M10 #Advanced Optimization Algorithms Research #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Fractional Differential Equations Solutions #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2509.11483
openalex publication_date 2025/09/15 · openalex created_date 2025/10/12 · openalex updated_date 2026/07/28
We analyze a second-order projection method for the incompressible Navier-Stokes equations on bounded Lipschitz domains. The scheme employs a Backward Differentiation Formula of order two (BDF2) for the time discretization, combined with conforming finite elements in space. Projection methods are widely used to enforce incompressibility, yet rigorous convergence results for possibly non-smooth solutions have so far been restricted to first-order schemes. We establish, for the first time, convergence (up to subsequence) of a second-order projection method to Leray-Hopf weak solutions under minimal assumptions on the data, namely u0 ∈ L2div(Ω) and f ∈ L2(0,T;L2div(Ω)). Our analysis relies on two ingredients: A discrete energy inequality providing uniform L∞(0,T;L2(Ω)) and L2(0,T;H10(Ω)) bounds for suitable interpolants of the discrete velocities, and a compactness argument combining Simon's theorem with refined time-continuity estimates. These tools overcome the difficulty that only the projected velocity satisfies an approximate divergence-free condition, while the intermediate velocity is controlled in space. We conclude that a subsequence of the approximations converges to a Leray-Hopf weak solution. This result provides the first rigorous convergence proof for a higher-order projection method under no additional assumptions on the solution beyond those following from the standard a priori energy estimate.