Pith Number

pith:4DIUCR2L

pith:2025:4DIUCR2LI54KAAAZY24GN6PNF6

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Optimal convergence rates for the finite element approximation of the Sobolev constant

Enrique Zuazua, Liviu I. Ignat

P1 finite elements achieve optimal convergence rates when approximating the Sobolev constant.

arxiv:2504.09637 v3 · 2025-04-13 · math.NA · cs.NA · math.CA

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Claims

C1strongest claim

We establish optimal convergence rates for the P1 finite element approximation of the Sobolev constant in arbitrary dimensions N≥2 and for Lebesgue exponents 1<p<N.

C2weakest assumption

The analysis relies on a refined study of the Sobolev deficit in suitable quasi-norms introduced and utilized in the context of finite element approximations of the p-Laplacian, together with sharp estimates for the finite element approximation of Sobolev minimizers.

C3one line summary

Optimal convergence rates are established for the P1 finite element approximation of the Sobolev constant in dimensions N≥2 for 1<p<N using refined Sobolev deficit analysis.

Formal links

1 machine-checked theorem link

Cited by

1 paper in Pith

Galerkin Approximation of the Fractional Sobolev Constant

Receipt and verification

First computed	2026-05-28T01:04:27.566629Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

e0d141474b4778a00019c6b866f9ed2f9585e501c7f615bb08ca08dbb5e01cce

Aliases

arxiv: 2504.09637 · arxiv_version: 2504.09637v3 · doi: 10.48550/arxiv.2504.09637 · pith_short_12: 4DIUCR2LI54K · pith_short_16: 4DIUCR2LI54KAAAZ · pith_short_8: 4DIUCR2L

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/4DIUCR2LI54KAAAZY24GN6PNF6 \
  | 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: e0d141474b4778a00019c6b866f9ed2f9585e501c7f615bb08ca08dbb5e01cce

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "50da0ed243d79ff7a157f3c04799670d8bb2c108d516abfb52d03b9620ed0c5a",
    "cross_cats_sorted": [
      "cs.NA",
      "math.CA"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NA",
    "submitted_at": "2025-04-13T16:22:05Z",
    "title_canon_sha256": "ea053251127860795a64422439a0287b33f727558b27103510620f252dbbb859"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2504.09637",
    "kind": "arxiv",
    "version": 3
  }
}