{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:2QSOQSADI7NDA6OLXPCFYOITCJ","short_pith_number":"pith:2QSOQSAD","canonical_record":{"source":{"id":"1110.5159","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-10-24T08:25:48Z","cross_cats_sorted":[],"title_canon_sha256":"fd6603fe88c573906d95b9fa57394552ce5a904fc755237d7ee73ffd07b0fe47","abstract_canon_sha256":"3914268b207679694b51dff6ab7705b4416f43ddc0ecc6c3d583593212e7900d"},"schema_version":"1.0"},"canonical_sha256":"d424e8480347da3079cbbbc45c39131250240c14e7ba6059ef16314c3c102688","source":{"kind":"arxiv","id":"1110.5159","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.5159","created_at":"2026-05-18T04:10:24Z"},{"alias_kind":"arxiv_version","alias_value":"1110.5159v1","created_at":"2026-05-18T04:10:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.5159","created_at":"2026-05-18T04:10:24Z"},{"alias_kind":"pith_short_12","alias_value":"2QSOQSADI7ND","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"2QSOQSADI7NDA6OL","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"2QSOQSAD","created_at":"2026-05-18T12:26:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:2QSOQSADI7NDA6OLXPCFYOITCJ","target":"record","payload":{"canonical_record":{"source":{"id":"1110.5159","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-10-24T08:25:48Z","cross_cats_sorted":[],"title_canon_sha256":"fd6603fe88c573906d95b9fa57394552ce5a904fc755237d7ee73ffd07b0fe47","abstract_canon_sha256":"3914268b207679694b51dff6ab7705b4416f43ddc0ecc6c3d583593212e7900d"},"schema_version":"1.0"},"canonical_sha256":"d424e8480347da3079cbbbc45c39131250240c14e7ba6059ef16314c3c102688","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:10:24.568599Z","signature_b64":"MQGp0jovReiuWAREIA2YRI8Ub7LvvoDy0mIICOrFPiIRsPVJAZmeBj9GcrRJ+aKHHhywOFVhy3l+Nm1fVBLJBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d424e8480347da3079cbbbc45c39131250240c14e7ba6059ef16314c3c102688","last_reissued_at":"2026-05-18T04:10:24.568149Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:10:24.568149Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1110.5159","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:10:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FCh/7iiQTz9yhmoWr9MdY/FI5ooXShk+qNfln9qnJDhvXZVfgNZIzfr160h8mwSYyHqxncIHETG5h/XVIvd5BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T04:02:55.652605Z"},"content_sha256":"9cf7c861decee6e796d1c3ae29cdf200aa02490518e9c4c611cd38d10a0ee995","schema_version":"1.0","event_id":"sha256:9cf7c861decee6e796d1c3ae29cdf200aa02490518e9c4c611cd38d10a0ee995"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:2QSOQSADI7NDA6OLXPCFYOITCJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Analysis of a multigrid preconditioner for Crouzeix-Raviart discretization of elliptic PDE with jump coefficient","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Yunrong Zhu","submitted_at":"2011-10-24T08:25:48Z","abstract_excerpt":"In this paper, we present a multigrid $V$-cycle preconditioner for the linear system arising from piecewise linear nonconforming Crouzeix-Raviart discretization of second order elliptic problems with jump coefficients. The preconditioner uses standard conforming subspaces as coarse spaces. We showed that the convergence rate of the multigrid $V$-cycle algorithm will deteriorate rapidly due to large jumps in coefficient. However, the preconditioned system has only a fixed number of small eigenvalues, which are deteriorated due to the large jump in coefficient, and the effective condition number"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.5159","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:10:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DRulHshaq6mgiR1XCDdNemZYmyjXGBDP4PKUz9klPdP6qtFWScOZmJ54Xc2AGSssm60zJ7ydCpcsIjzNoscVDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T04:02:55.652951Z"},"content_sha256":"6f3af97e76e77f4d467bd7739dfb6389ff790cf2a6d3c644edc9e68842c42587","schema_version":"1.0","event_id":"sha256:6f3af97e76e77f4d467bd7739dfb6389ff790cf2a6d3c644edc9e68842c42587"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2QSOQSADI7NDA6OLXPCFYOITCJ/bundle.json","state_url":"https://pith.science/pith/2QSOQSADI7NDA6OLXPCFYOITCJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2QSOQSADI7NDA6OLXPCFYOITCJ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-20T04:02:55Z","links":{"resolver":"https://pith.science/pith/2QSOQSADI7NDA6OLXPCFYOITCJ","bundle":"https://pith.science/pith/2QSOQSADI7NDA6OLXPCFYOITCJ/bundle.json","state":"https://pith.science/pith/2QSOQSADI7NDA6OLXPCFYOITCJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2QSOQSADI7NDA6OLXPCFYOITCJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:2QSOQSADI7NDA6OLXPCFYOITCJ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"3914268b207679694b51dff6ab7705b4416f43ddc0ecc6c3d583593212e7900d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-10-24T08:25:48Z","title_canon_sha256":"fd6603fe88c573906d95b9fa57394552ce5a904fc755237d7ee73ffd07b0fe47"},"schema_version":"1.0","source":{"id":"1110.5159","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.5159","created_at":"2026-05-18T04:10:24Z"},{"alias_kind":"arxiv_version","alias_value":"1110.5159v1","created_at":"2026-05-18T04:10:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.5159","created_at":"2026-05-18T04:10:24Z"},{"alias_kind":"pith_short_12","alias_value":"2QSOQSADI7ND","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"2QSOQSADI7NDA6OL","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"2QSOQSAD","created_at":"2026-05-18T12:26:18Z"}],"graph_snapshots":[{"event_id":"sha256:6f3af97e76e77f4d467bd7739dfb6389ff790cf2a6d3c644edc9e68842c42587","target":"graph","created_at":"2026-05-18T04:10:24Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this paper, we present a multigrid $V$-cycle preconditioner for the linear system arising from piecewise linear nonconforming Crouzeix-Raviart discretization of second order elliptic problems with jump coefficients. The preconditioner uses standard conforming subspaces as coarse spaces. We showed that the convergence rate of the multigrid $V$-cycle algorithm will deteriorate rapidly due to large jumps in coefficient. However, the preconditioned system has only a fixed number of small eigenvalues, which are deteriorated due to the large jump in coefficient, and the effective condition number","authors_text":"Yunrong Zhu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-10-24T08:25:48Z","title":"Analysis of a multigrid preconditioner for Crouzeix-Raviart discretization of elliptic PDE with jump coefficient"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.5159","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:9cf7c861decee6e796d1c3ae29cdf200aa02490518e9c4c611cd38d10a0ee995","target":"record","created_at":"2026-05-18T04:10:24Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"3914268b207679694b51dff6ab7705b4416f43ddc0ecc6c3d583593212e7900d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-10-24T08:25:48Z","title_canon_sha256":"fd6603fe88c573906d95b9fa57394552ce5a904fc755237d7ee73ffd07b0fe47"},"schema_version":"1.0","source":{"id":"1110.5159","kind":"arxiv","version":1}},"canonical_sha256":"d424e8480347da3079cbbbc45c39131250240c14e7ba6059ef16314c3c102688","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d424e8480347da3079cbbbc45c39131250240c14e7ba6059ef16314c3c102688","first_computed_at":"2026-05-18T04:10:24.568149Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:10:24.568149Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MQGp0jovReiuWAREIA2YRI8Ub7LvvoDy0mIICOrFPiIRsPVJAZmeBj9GcrRJ+aKHHhywOFVhy3l+Nm1fVBLJBg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:10:24.568599Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.5159","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9cf7c861decee6e796d1c3ae29cdf200aa02490518e9c4c611cd38d10a0ee995","sha256:6f3af97e76e77f4d467bd7739dfb6389ff790cf2a6d3c644edc9e68842c42587"],"state_sha256":"7d907f44eb9547a56d701ed01d9267dc620d0df17fd714c6bd696341e8ea623a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lpHAqt2fqL9kj69U+sPehOF7VBW9PobOFv28xizI2Mc0YFM6QHh+TOFBMcRlbfia/VNRXJyn12OtFS+qvhtSAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T04:02:55.655072Z","bundle_sha256":"061c62908e5be39d437b491be344724f0062172c81aa3c350fa07795e415e860"}}