{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2023:SCY2NNBOMV3HMCAPMUPMIHQCV3","short_pith_number":"pith:SCY2NNBO","schema_version":"1.0","canonical_sha256":"90b1a6b42e657676080f651ec41e02aec1e2da0f02ddb235675ad31de4c4a219","source":{"kind":"arxiv","id":"2305.03575","version":1},"attestation_state":"computed","paper":{"title":"Pointwise gradient estimate of the ritz projection","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Abner J. Salgado, Julian Rolfes, Lars Diening","submitted_at":"2023-05-05T14:30:47Z","abstract_excerpt":"Let $\\Omega \\subset \\mathbb{R}^n$ be a convex polytope ($n \\leq 3$). The Ritz projection is the best approximation, in the $W^{1,2}_0$-norm, to a given function in a finite element space. When such finite element spaces are constructed on the basis of quasiuniform triangulations, we show a pointwise estimate on the Ritz projection. Namely, that the gradient at any point in $\\Omega$ is controlled by the Hardy--Littlewood maximal function of the gradient of the original function at the same point. From this estimate, the stability of the Ritz projection on a wide range of spaces that are of inte"},"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":"2305.03575","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2023-05-05T14:30:47Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"8290a0f9f3963e0214176f16b6258f2376e4cdd984a411a2283bb3ede539ad11","abstract_canon_sha256":"2e23159e7b8ee0887497f7a324ac58ef3b754fbada85cb5737458f4928b81aaa"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:07:24.742279Z","signature_b64":"dFuE8CqeoCUrPSQt+vPT7wn3+CoHhFxAJNpLiTONWo4cWdpKEaIA5L1bmv7Ay6A/IC+hLs564FeNDi/+ajhGDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"90b1a6b42e657676080f651ec41e02aec1e2da0f02ddb235675ad31de4c4a219","last_reissued_at":"2026-07-05T06:07:24.741892Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:07:24.741892Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Pointwise gradient estimate of the ritz projection","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Abner J. Salgado, Julian Rolfes, Lars Diening","submitted_at":"2023-05-05T14:30:47Z","abstract_excerpt":"Let $\\Omega \\subset \\mathbb{R}^n$ be a convex polytope ($n \\leq 3$). The Ritz projection is the best approximation, in the $W^{1,2}_0$-norm, to a given function in a finite element space. When such finite element spaces are constructed on the basis of quasiuniform triangulations, we show a pointwise estimate on the Ritz projection. Namely, that the gradient at any point in $\\Omega$ is controlled by the Hardy--Littlewood maximal function of the gradient of the original function at the same point. From this estimate, the stability of the Ritz projection on a wide range of spaces that are of inte"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2305.03575","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2305.03575/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2305.03575","created_at":"2026-07-05T06:07:24.741948+00:00"},{"alias_kind":"arxiv_version","alias_value":"2305.03575v1","created_at":"2026-07-05T06:07:24.741948+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2305.03575","created_at":"2026-07-05T06:07:24.741948+00:00"},{"alias_kind":"pith_short_12","alias_value":"SCY2NNBOMV3H","created_at":"2026-07-05T06:07:24.741948+00:00"},{"alias_kind":"pith_short_16","alias_value":"SCY2NNBOMV3HMCAP","created_at":"2026-07-05T06:07:24.741948+00:00"},{"alias_kind":"pith_short_8","alias_value":"SCY2NNBO","created_at":"2026-07-05T06:07:24.741948+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2605.20367","citing_title":"A Penalty-Free Asymmetric Nitsche's Method for Edge Elements","ref_index":213,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/SCY2NNBOMV3HMCAPMUPMIHQCV3","json":"https://pith.science/pith/SCY2NNBOMV3HMCAPMUPMIHQCV3.json","graph_json":"https://pith.science/api/pith-number/SCY2NNBOMV3HMCAPMUPMIHQCV3/graph.json","events_json":"https://pith.science/api/pith-number/SCY2NNBOMV3HMCAPMUPMIHQCV3/events.json","paper":"https://pith.science/paper/SCY2NNBO"},"agent_actions":{"view_html":"https://pith.science/pith/SCY2NNBOMV3HMCAPMUPMIHQCV3","download_json":"https://pith.science/pith/SCY2NNBOMV3HMCAPMUPMIHQCV3.json","view_paper":"https://pith.science/paper/SCY2NNBO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2305.03575&json=true","fetch_graph":"https://pith.science/api/pith-number/SCY2NNBOMV3HMCAPMUPMIHQCV3/graph.json","fetch_events":"https://pith.science/api/pith-number/SCY2NNBOMV3HMCAPMUPMIHQCV3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SCY2NNBOMV3HMCAPMUPMIHQCV3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SCY2NNBOMV3HMCAPMUPMIHQCV3/action/storage_attestation","attest_author":"https://pith.science/pith/SCY2NNBOMV3HMCAPMUPMIHQCV3/action/author_attestation","sign_citation":"https://pith.science/pith/SCY2NNBOMV3HMCAPMUPMIHQCV3/action/citation_signature","submit_replication":"https://pith.science/pith/SCY2NNBOMV3HMCAPMUPMIHQCV3/action/replication_record"}},"created_at":"2026-07-05T06:07:24.741948+00:00","updated_at":"2026-07-05T06:07:24.741948+00:00"}