Scott–Vogelius–Nitsche on a polygonally approximated boundary: convergence, the missing pressure traction, and the penalty threshold

We analyse the Scott–Vogelius–Nitsche method of Gjerde and Scott for two-dimensional Stokes flow whose no-slip curve is replaced by an inscribed polygon: continuous P4 velocities that are exactly divergence-free, with Dirichlet data imposed weakly on the polygon by Nitsche's method. This answers Ridgway Scott's zero-gradient prize question (PPL 115), which asks whether the H¹ error behaves like h_Γ^{3/2} + h_Ω^k for general Stokes data and, if not, why not. For the method as printed, the answer is no as soon as the pressure is not constant on the boundary. The printed Nitsche form omits the traction −pn. We prove that the method converges, at the sharp rate h/μ in the H¹ seminorm and at the exact rate (h/μ)^{1/2} in the mesh-dependent energy norm. To leading order the error is an explicit field: the discrete velocity leaks through the wall with normal velocity (h/μ)(p − p̄), with constant exactly 1 in the slip and energy norms. The H¹ rate is 1 rather than 1/2 only because incompressibility turns the normal part of the missing traction into a tangential derivative. The naive pressure-consistent correction is singular. A mean-corrected variant is well posed and attains h_Γ^{3/2} + h_Ω^4 for general data, using the Guzmán–Scott inf-sup constant made uniform over the perturbed domains. We also show that the Nitsche penalty must scale like ρ = h_Ω / min|e|, with necessity certified in exact rational arithmetic. Computations with up to N = 256 boundary chords confirm every rate and constant. They include a pure-pressure test that isolates the traction response exactly, and a penalty study in which the threshold is linear in ρ with slope about 20. Code, raw results and logs: https://github.com/jaideepsaipadhi/zero-gradient-prize