Convergence varying for Newtons Method

Dear All, Anyone has working experience with the NS method mentioned in the ( Navier-Stokes).

My solution is not converging for this approach.

I used Newton method to solve NS equations (2-D driven-cavity flow) in FreeFem++ (old FreeFEM), but it was 4 years ago. If your coefficient matrix and right-hand side assembled are correct, generally, the nonlinear convergence problem may be caused by a very large Reynolds number.
As suggested in the example, you can use the strategy of continuous vanishing viscosity, and you can add some stabilization methods such as SUPG/PSPG, GLS etc. (practically, it can simulate 0.5~1 Million Reynolds number, but unfortunately, it is not a physical solution in this case). Algebraically, you can also refer to Augmented Lagrangian preconditioner in GitHub - prj-/moulin2019al: Augmented Lagrangian Preconditioner for Hydrodynamic Stability Analysis by Pierre, Johann, Olivier.
In addition, using PETSc/SNES for nonlinear iteration is also recommended because more robust.

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Thanks, @zhaog6 Sir. It is really helpful.