Finite Element Analysis for the Stationary Navier–Stokes Equations with Mixed Boundary Conditions

This paper studies the stationary incompressible Navier-Stokes equations with mixed boundary conditions using a velocity-pressure finite element formulation. We first establish a variational framework and prove existence of solutions under suitable regularity assumptions, followed by a Galerkin discretization with error estimates. Three iterative algorithms (the Stokes, Newton, and Oseen schemes) are then analyzed, with stability conditions and error bounds derived for each. Numerical experiments confirm the theoretical results: all methods achieve second-order convergence for velocity and pressure. Among the three schemes, the Newton iteration is the most efficient in terms of computational time, while the Oseen iteration exhibits the strongest robustness with respect to decreasing viscosity coefficients.