Equal order stabilized finite elements with Nitsche for stationary Navier-Stokes problem with slip boundary conditions : a priori and a posteriori error analysis

TL;DR

This paper develops a stabilized finite element method with Nitsche's technique for the stationary Navier-Stokes equations with slip boundary conditions, providing theoretical analysis and numerical validation.

Contribution

It extends equal-order stabilized schemes to slip boundary conditions using Nitsche's method, with proven well-posedness, optimal convergence, and reliable a posteriori error estimators.

Findings

Robust formulation for slip boundary conditions.

Optimal convergence rates demonstrated.

Effective residual-based error estimators validated.

Abstract

In this work, we extend the equal-order stabilized scheme discussed in [Franca et al., Comput. Methods Appl. Mech. Engrg. 99 (1992) 209-233] to accommodate slip (i.e., Navier) boundary conditions for the stationary Navier-Stokes equations. Our analysis presents a robust formulation for implementing slip boundary conditions using Nitsche's method on arbitrarily complex boundaries. The well-posedness of the discrete problem is established under mild assumptions together with optimal convergence rates for the approximation error. Furthermore, we establish the efficiency and reliability of residual-based a posteriori error estimators for the stationary discrete problem. Several well-known numerical tests validate our theoretical findings. The proposed method fits naturally within the framework of finite element implementation, offering both accuracy and enhanced flexibility in the selection of finite element pairs.