{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:RFEJ27EBK3PZQELUQZ2H435WPR","short_pith_number":"pith:RFEJ27EB","schema_version":"1.0","canonical_sha256":"89489d7c8156df98117486747e6fb67c49e0662b31e2d4d0a59ce038914d3462","source":{"kind":"arxiv","id":"2501.15976","version":4},"attestation_state":"computed","paper":{"title":"Theory of two-level Schwarz preconditioners with piecewise-polynomial coarse spaces for the high-frequency Helmholtz equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Euan A. Spence, Ivan G. Graham","submitted_at":"2025-01-27T11:49:59Z","abstract_excerpt":"We analyse the classic two-level additive Schwarz domain-decomposition GMRES preconditioner for finite-element discretisations of the Helmholtz equation with large wavenumber $k$, where both the fine and coarse spaces consist of piecewise polynomials with polynomial degree increasing like $\\log k$.\n  We exhibit choices of these fine and coarse spaces such that -- up to factors of $\\log k$ -- both are pollution free (with the ratio of the coarse-space dimension to the fine-space dimension arbitrarily small), the number of degrees of freedom per subdomain is constant, and the number of GMRES ite"},"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":"2501.15976","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2025-01-27T11:49:59Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"b3c972c0eb3dea8016105d3b145a6c9f5ca3ec39be509cbaa9eecb01075e8333","abstract_canon_sha256":"a5a76b17f8062fe0cefe102a3ef2048b8b9979668075677796a050dee25dd758"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:12:44.920098Z","signature_b64":"Y7VhBbaHQp5+FRmm3f3dwL3Yz2pD4FT+c7dcVHqGCc5P9DxTORX5Pka0ZiBD92eFccYZhQeXB3Al6aAGXpMeDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"89489d7c8156df98117486747e6fb67c49e0662b31e2d4d0a59ce038914d3462","last_reissued_at":"2026-06-19T16:12:44.919630Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:12:44.919630Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Theory of two-level Schwarz preconditioners with piecewise-polynomial coarse spaces for the high-frequency Helmholtz equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Euan A. Spence, Ivan G. Graham","submitted_at":"2025-01-27T11:49:59Z","abstract_excerpt":"We analyse the classic two-level additive Schwarz domain-decomposition GMRES preconditioner for finite-element discretisations of the Helmholtz equation with large wavenumber $k$, where both the fine and coarse spaces consist of piecewise polynomials with polynomial degree increasing like $\\log k$.\n  We exhibit choices of these fine and coarse spaces such that -- up to factors of $\\log k$ -- both are pollution free (with the ratio of the coarse-space dimension to the fine-space dimension arbitrarily small), the number of degrees of freedom per subdomain is constant, and the number of GMRES ite"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.15976","kind":"arxiv","version":4},"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/2501.15976/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":"2501.15976","created_at":"2026-06-19T16:12:44.919687+00:00"},{"alias_kind":"arxiv_version","alias_value":"2501.15976v4","created_at":"2026-06-19T16:12:44.919687+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2501.15976","created_at":"2026-06-19T16:12:44.919687+00:00"},{"alias_kind":"pith_short_12","alias_value":"RFEJ27EBK3PZ","created_at":"2026-06-19T16:12:44.919687+00:00"},{"alias_kind":"pith_short_16","alias_value":"RFEJ27EBK3PZQELU","created_at":"2026-06-19T16:12:44.919687+00:00"},{"alias_kind":"pith_short_8","alias_value":"RFEJ27EB","created_at":"2026-06-19T16:12:44.919687+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/RFEJ27EBK3PZQELUQZ2H435WPR","json":"https://pith.science/pith/RFEJ27EBK3PZQELUQZ2H435WPR.json","graph_json":"https://pith.science/api/pith-number/RFEJ27EBK3PZQELUQZ2H435WPR/graph.json","events_json":"https://pith.science/api/pith-number/RFEJ27EBK3PZQELUQZ2H435WPR/events.json","paper":"https://pith.science/paper/RFEJ27EB"},"agent_actions":{"view_html":"https://pith.science/pith/RFEJ27EBK3PZQELUQZ2H435WPR","download_json":"https://pith.science/pith/RFEJ27EBK3PZQELUQZ2H435WPR.json","view_paper":"https://pith.science/paper/RFEJ27EB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2501.15976&json=true","fetch_graph":"https://pith.science/api/pith-number/RFEJ27EBK3PZQELUQZ2H435WPR/graph.json","fetch_events":"https://pith.science/api/pith-number/RFEJ27EBK3PZQELUQZ2H435WPR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RFEJ27EBK3PZQELUQZ2H435WPR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RFEJ27EBK3PZQELUQZ2H435WPR/action/storage_attestation","attest_author":"https://pith.science/pith/RFEJ27EBK3PZQELUQZ2H435WPR/action/author_attestation","sign_citation":"https://pith.science/pith/RFEJ27EBK3PZQELUQZ2H435WPR/action/citation_signature","submit_replication":"https://pith.science/pith/RFEJ27EBK3PZQELUQZ2H435WPR/action/replication_record"}},"created_at":"2026-06-19T16:12:44.919687+00:00","updated_at":"2026-06-19T16:12:44.919687+00:00"}