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

Abstract

Extended version (v3). We analyse the Scott–Vogelius–Nitsche method of Gjerde and Scott for two-dimensional Stokes flow whose curved no-slip wall is replaced by an inscribed polygon, answering Scott's zero-gradient prize question (PPL 115): does the error behave like h_Γ^{3/2} + h^k, and if not, why not? For the method as printed, the answer is no whenever the pressure varies along the wall. The Nitsche form omits the pressure traction, and the discrete velocity leaks through the wall with normal velocity (h/μ)(p − p̄). We prove the leak law with exact constants and the resulting rates. The mean-free traction correction (a device due to Frachon–Nilsson–Zahedi; the analysis is new here) attains Scott's rate h_Γ^{3/2} + h^k for general Stokes data and every k ≥ 4. We also prove the matching lower bound, so the rate is sharp. Further results: the penalty threshold μ ∝ h/min|e|; strong imposition and exactly when the zero-gradient locking occurs; pressure robustness; L² estimates; convergence uniform in the penalty; Navier–Stokes on nonsingular branches; Clough–Tocher refinements and general curved domains. This 156-page version contains all proofs and is the supplement to a shorter journal version. Code, certificates and logs: https://github.com/jaideepsaipadhi/zero-gradient-prize