{"paper":{"title":"Sharp inf-sup estimate for the Stokes equation in tight domains with periodic pillars and some numerical implications","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The inf-sup constant for the Stokes equations in tight domains with periodic pillars degrades exactly as the inverse of pillar density m.","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Jinchao Xu, Qi Xin, Shihua Gong","submitted_at":"2026-04-14T12:13:06Z","abstract_excerpt":"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 de"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we prove that the continuous Ladyzhenskaya-Babuška-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 causes a severe a priori error amplification and extreme ill-conditioning in Schur complement of the saddle point system.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The analysis assumes periodic pillar geometries within a generalized lattice framework for tight domains, with the inf-sup degradation being exactly proportional to m^{-1} without other geometric factors dominating the constant.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The inf-sup constant for Stokes flow with periodic pillars decays as m^{-1}, causing ill-conditioning that a parameter-free adaptive Augmented Lagrangian method overcomes in numerical tests.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The inf-sup constant for the Stokes equations in tight domains with periodic pillars degrades exactly as the inverse of pillar density m.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"77cdc0af5017d8bf62e696bd4b26eb039b4f856e8a09fe689f402dca6b7204da"},"source":{"id":"2604.12643","kind":"arxiv","version":2},"verdict":{"id":"203bd6e0-361a-410d-a29b-3fe19a45ca60","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T15:11:29.783893Z","strongest_claim":"we prove that the continuous Ladyzhenskaya-Babuška-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 causes a severe a priori error amplification and extreme ill-conditioning in Schur complement of the saddle point system.","one_line_summary":"The inf-sup constant for Stokes flow with periodic pillars decays as m^{-1}, causing ill-conditioning that a parameter-free adaptive Augmented Lagrangian method overcomes in numerical tests.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The analysis assumes periodic pillar geometries within a generalized lattice framework for tight domains, with the inf-sup degradation being exactly proportional to m^{-1} without other geometric factors dominating the constant.","pith_extraction_headline":"The inf-sup constant for the Stokes equations in tight domains with periodic pillars degrades exactly as the inverse of pillar density m."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.12643/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}