Sharp inf-sup estimate for the Stokes equation in tight domains with periodic pillars and some numerical implications

The predictive simulation of fluid dynamics in densely packed microfluidic devices, such as Deterministic Lateral Displacement (DLD) arrays, stagnates with standard iterative solvers. We show that this failure is not algorithmic but rooted in the pre-asymptotic degradation of the pressure-velocity coupling stability. For periodic pillar geometries in a generalized lattice framework, we prove that the continuous Ladyzhenskaya-Babu\v{s}ka-Brezzi (LBB) condition, also called the inf-sup constant, deteriorates exactly as $m^{-1}$ up to a positive multiplicative constant, where $m$ is the pillar density (the number of pillars per unit length). This induces a priori error amplification proportional to $m$ and a pressure Schur complement condition number scaling as $\mathcal{O}(m^2)$. To overcome this theoretical limit, we propose a parameter-free, adaptively scaled Augmented Lagrangian (AL) stabilization strategy with penalty $\gamma \propto m^2$. Numerical experiments on both standard square and asymmetric DLD arrays validate the theoretical bounds: the AL method reduces outer FGMRES iterations from 437 to 22 on a 1.85M-DoF square array and from 687 to 24 on a 1.77M-DoF DLD array.