Finite element approximation of the stationary Navier-Stokes problem with non-smooth data

The aim of this work is to analyze the finite element approximation of the two-dimensional stationary Navier-Stokes equations with non-smooth Dirichlet boundary data. The discrete approximation is obtained by considering the Navier-Stokes system with a regularized boundary solution. Based on the existence of the very weak solution for the Navier-Stokes system with L2 boundary data, and a suitable decomposition of this solution, we obtain a priori error estimates between the approximation of the Navier-Stokes system with non-smooth data and the finite element solution of the associated regularized problem. These estimates allow us to conclude that our approach converges with optimal order.

💡 Research Summary

The paper addresses the numerical approximation of the two‑dimensional stationary Navier–Stokes equations when the prescribed Dirichlet boundary data belong only to L²(Γ), a regularity level that is insufficient for the classical variational formulation. The authors develop a rigorous finite‑element convergence theory that works under this low‑regularity setting.

First, they introduce the concept of a “very weak solution” (u,p) defined on the test spaces X = {φ ∈ W^{2,4/3}(Ω)∩W^{1,4/3}_0(Ω) : div φ = 0} and M = {q ∈ W^{1,4/3}(Ω) : ∫_Ω q = 0}. In this framework the velocity belongs to L⁴(Ω) and the pressure to W^{−1,4}(Ω), which is enough to give meaning to the nonlinear convective term. Existence and uniqueness of such a solution are proved by adapting results from Marušić‑Paloka (reference