{"paper":{"title":"Optimal convergence rates for the finite element approximation of the Sobolev constant","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"P1 finite elements achieve optimal convergence rates when approximating the Sobolev constant.","cross_cats":["cs.NA","math.CA"],"primary_cat":"math.NA","authors_text":"Enrique Zuazua, Liviu I. Ignat","submitted_at":"2025-04-13T16:22:05Z","abstract_excerpt":"We establish optimal convergence rates for the continuous piecewise affine finite element approximation of the Sobolev constant in arbitrary dimensions N\\geq 2 and for Lebesgue exponents 1<p<N. Our analysis relies on a refined study of the Sobolev deficit in suitable quasi-norms, which have been introduced and utilized in the context of finite element approximations of the p-Laplacian. The proof further involves sharp estimates for the finite element approximation of Sobolev minimizers."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"P1 finite elements achieve optimal convergence rates when approximating the Sobolev constant.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"9761ba2d108e4194ee04635e923f035b957cbc8a53c231a8cb5c84e55c41554c"},"source":{"id":"2504.09637","kind":"arxiv","version":3},"verdict":{"id":"d93f212c-b710-4061-b374-6f782e3f1b89","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-22T20:26:38.468689Z","strongest_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.","one_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.","pipeline_version":"pith-pipeline@v0.9.0","weakest_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.","pith_extraction_headline":"P1 finite elements achieve optimal convergence rates when approximating the Sobolev constant."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2504.09637/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":1,"snapshot_sha256":"ee54a0527845d094d2c28b8a63d5cb9e321bae2ea2443de65700215d54b822b5"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}