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

TL;DR

This work addresses finite element approximation of the stationary Navier–Stokes equations in two dimensions with non-smooth Dirichlet boundary data. The authors regularize the boundary data and analyze the resulting regularized problem through a very weak-solution framework, establishing a decomposition $u = u_\varepsilon + v_\varepsilon$ and deriving error bounds that relate the non-smooth problem to its regularized counterpart. They prove existence (and a small-data uniqueness) for very weak solutions and obtain a priori FE error estimates that scale as $\|u-u_h\|_{L^4(\Omega)} \le C h^{\min\{r,s\}}$, with $r$ the FE interpolation order and $s$ the boundary-data-approximation rate. Numerical experiments on the lid-driven cavity validate the theory across multiple FE discretizations and Newton iterations, demonstrating convergence consistent with the predicted rates. Overall, the paper provides a rigorous, practical framework for reliable FE simulations of incompressible flows with low-regularity boundary data.

Abstract

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.

Figures (1)

• Figure 1: Convergence order analysis.

Theorems & Definitions (25)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Remark 2.4
• Remark 2.5
• Lemma 2.6
• proof