{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:4UDPGC5MURY3LXUI4UC6B4OEA4","short_pith_number":"pith:4UDPGC5M","schema_version":"1.0","canonical_sha256":"e506f30baca471b5de88e505e0f1c40700c083359b95e9c794de204fadef1a0e","source":{"kind":"arxiv","id":"1112.3705","version":1},"attestation_state":"computed","paper":{"title":"Superconvergence of the $Q_{k+1,k}$-$Q_{k,k+1}$ divergence-free finite element","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Shangyou Zhang, Yunqing Huang","submitted_at":"2011-12-16T03:30:38Z","abstract_excerpt":"By the standard theory, the stable $Q_{k+1,k}$-$Q_{k,k+1}/Q_{k}^{dc'}$ divergence-free element converges with the optimal order of approximation for the Stokes equations, but only order $k$ for the velocity in $H^1$-norm and the pressure in $L^2$-norm. This is due to one polynomial degree less in $y$ direction for the first component of velocity, a $Q_{k+1,k}$ polynomial. In this manuscript, we will show a superconvergence of the divergence free element that the order of convergence is truly $k+1$, for both velocity and pressure. Numerical tests are provided confirming the sharpness of the the"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1112.3705","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-12-16T03:30:38Z","cross_cats_sorted":[],"title_canon_sha256":"95327bb073ef9af23baac90072283360ac91141b2e17089a858f827a67ba9d2c","abstract_canon_sha256":"8e781f3f3d9062f0d9c33560b2ca2df84bbb0bd8f502e3ca4773fb50612ba799"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:06:09.073499Z","signature_b64":"BXgnXhpCYwVLTpuzjvn6AIGQRoNAlQzI0yXxwKF/wuBZwfpHIj9wuiUIGW6eG16z629Seb71LQFHnj3Lsc0XAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e506f30baca471b5de88e505e0f1c40700c083359b95e9c794de204fadef1a0e","last_reissued_at":"2026-05-18T04:06:09.072884Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:06:09.072884Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Superconvergence of the $Q_{k+1,k}$-$Q_{k,k+1}$ divergence-free finite element","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Shangyou Zhang, Yunqing Huang","submitted_at":"2011-12-16T03:30:38Z","abstract_excerpt":"By the standard theory, the stable $Q_{k+1,k}$-$Q_{k,k+1}/Q_{k}^{dc'}$ divergence-free element converges with the optimal order of approximation for the Stokes equations, but only order $k$ for the velocity in $H^1$-norm and the pressure in $L^2$-norm. This is due to one polynomial degree less in $y$ direction for the first component of velocity, a $Q_{k+1,k}$ polynomial. In this manuscript, we will show a superconvergence of the divergence free element that the order of convergence is truly $k+1$, for both velocity and pressure. Numerical tests are provided confirming the sharpness of the the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.3705","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1112.3705","created_at":"2026-05-18T04:06:09.072976+00:00"},{"alias_kind":"arxiv_version","alias_value":"1112.3705v1","created_at":"2026-05-18T04:06:09.072976+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.3705","created_at":"2026-05-18T04:06:09.072976+00:00"},{"alias_kind":"pith_short_12","alias_value":"4UDPGC5MURY3","created_at":"2026-05-18T12:26:20.644004+00:00"},{"alias_kind":"pith_short_16","alias_value":"4UDPGC5MURY3LXUI","created_at":"2026-05-18T12:26:20.644004+00:00"},{"alias_kind":"pith_short_8","alias_value":"4UDPGC5M","created_at":"2026-05-18T12:26:20.644004+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/4UDPGC5MURY3LXUI4UC6B4OEA4","json":"https://pith.science/pith/4UDPGC5MURY3LXUI4UC6B4OEA4.json","graph_json":"https://pith.science/api/pith-number/4UDPGC5MURY3LXUI4UC6B4OEA4/graph.json","events_json":"https://pith.science/api/pith-number/4UDPGC5MURY3LXUI4UC6B4OEA4/events.json","paper":"https://pith.science/paper/4UDPGC5M"},"agent_actions":{"view_html":"https://pith.science/pith/4UDPGC5MURY3LXUI4UC6B4OEA4","download_json":"https://pith.science/pith/4UDPGC5MURY3LXUI4UC6B4OEA4.json","view_paper":"https://pith.science/paper/4UDPGC5M","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1112.3705&json=true","fetch_graph":"https://pith.science/api/pith-number/4UDPGC5MURY3LXUI4UC6B4OEA4/graph.json","fetch_events":"https://pith.science/api/pith-number/4UDPGC5MURY3LXUI4UC6B4OEA4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4UDPGC5MURY3LXUI4UC6B4OEA4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4UDPGC5MURY3LXUI4UC6B4OEA4/action/storage_attestation","attest_author":"https://pith.science/pith/4UDPGC5MURY3LXUI4UC6B4OEA4/action/author_attestation","sign_citation":"https://pith.science/pith/4UDPGC5MURY3LXUI4UC6B4OEA4/action/citation_signature","submit_replication":"https://pith.science/pith/4UDPGC5MURY3LXUI4UC6B4OEA4/action/replication_record"}},"created_at":"2026-05-18T04:06:09.072976+00:00","updated_at":"2026-05-18T04:06:09.072976+00:00"}