Pith Number

pith:XCOJVPQH

pith:2026:XCOJVPQH36YLHNBEWWEYVBB7OX

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

Jinchao Xu, Qi Xin, Shihua Gong

The inf-sup constant for the Stokes equations in tight domains with periodic pillars degrades exactly as the inverse of pillar density m.

arxiv:2604.12643 v2 · 2026-04-14 · math.NA · cs.NA

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Claims

C1strongest 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.

C2weakest 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.

C3one 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.

Receipt and verification

First computed	2026-05-26T01:03:30.076595Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

b89c9abe07dfb0b3b424b5898a843f75face00ce0687876f2d1f21e914c4ec07

Aliases

arxiv: 2604.12643 · arxiv_version: 2604.12643v2 · doi: 10.48550/arxiv.2604.12643 · pith_short_12: XCOJVPQH36YL · pith_short_16: XCOJVPQH36YLHNBE · pith_short_8: XCOJVPQH

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/XCOJVPQH36YLHNBEWWEYVBB7OX \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: b89c9abe07dfb0b3b424b5898a843f75face00ce0687876f2d1f21e914c4ec07

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "63356c360743810673cd8dbbb1c1e9331d1024ccb134cc12a57f4ec876b5207d",
    "cross_cats_sorted": [
      "cs.NA"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.NA",
    "submitted_at": "2026-04-14T12:13:06Z",
    "title_canon_sha256": "3aaa45b63ab2d27e0e29798da819223e60f400dfd7def0ade3532649d4b3b657"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.12643",
    "kind": "arxiv",
    "version": 2
  }
}