Uniform Inf-Sup Norm Equivalence and Robust Operator Preconditioning for Stokes Flow in tight domains with Periodic Pillars
Many microfluidic and porous-media computations reduce to the same core task: solving a Stokes saddle-point system on a domain perforated by a dense periodic array of pillars, as in deterministic lateral displacement (DLD) particle sorters. After rescaling the device to unit size, the geometry is controlled by a single dimensionless parameter $m$---the number of pillars across the device, equal to the inverse period. In realistic devices $m$ reaches the hundreds or thousands, and as it grows the Stokes inf-sup constant decays like $m^{-1}$, the pressure Schur complement becomes severely ill-conditioned, and standard block solvers slow down in proportion to the pillar density. We remove this bottleneck by identifying the pressure norm that the divergence operator induces on such geometries. For periodic pillar arrays in the proportional-hole regime, we prove that this inf-sup norm is uniformly equivalent to the $L^2+\sigma_\epsilon H^1$ $K$-functional norm at the pore scale $\sigma_\epsilon\asymp\epsilon$, with constants independent of the period, the pillar count $m$, and the mesh size $h$. The equivalence identifies the perforated Stokes problem with a Brinkman problem at a homogenized permeability, and its Riesz map reduces to a pressure-mass inverse plus a scaled stiffness inverse. Combined with operator preconditioning, this yields a block preconditioner built from standard algebraic-multigrid solves whose iteration count is essentially independent of both mesh size and pillar density. Two-dimensional Taylor--Hood experiments confirm the predicted robustness in mesh refinement, pillar density, geometric scale, close packing, and time step.