Discrete ε-Regularity for Block-Structured Navier-Stokes Discretizations

We develop a discrete ε-regularity criterion for finite difference discretizations of the incompressible Navier-Stokes equations on periodic domains. The approach uses a block-structured computational framework with instrumented measurements over dyadic space-time cylinders. We prove a monotonicity inequality for an enhanced energy functional that controls kinetic energy, interface fluxes, divergence, vorticity, and pressure gradients. This leads to a quantitative regularity criterion: if scaled densities ofenstrophy, interface flux, and projection energy drop remain below an explicit threshold ε*, then thesolution exhibits Hölder continuity on smaller cylinders. Numerical experiments on 2D periodic flows demonstrate strong correlation between the combined metric and true regularity measures, with agreement rates exceeding 85% for identifying regular regions.