{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:DOLARGFFFCDOQKFZWZZZL4RRZP","short_pith_number":"pith:DOLARGFF","canonical_record":{"source":{"id":"1601.05680","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-01-21T15:37:54Z","cross_cats_sorted":[],"title_canon_sha256":"0006d09ed6d64f8621e4d8d385bf5f55684f12963570c32a9377a4abba2c662e","abstract_canon_sha256":"8397be97370ffc8dc4ef50a17d3047743bd10c9d978150c72edd5779b363ec74"},"schema_version":"1.0"},"canonical_sha256":"1b960898a52886e828b9b67395f231cbc839b4b0126573c4fb5cf1be47f5d217","source":{"kind":"arxiv","id":"1601.05680","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.05680","created_at":"2026-05-18T01:22:13Z"},{"alias_kind":"arxiv_version","alias_value":"1601.05680v1","created_at":"2026-05-18T01:22:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.05680","created_at":"2026-05-18T01:22:13Z"},{"alias_kind":"pith_short_12","alias_value":"DOLARGFFFCDO","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_16","alias_value":"DOLARGFFFCDOQKFZ","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_8","alias_value":"DOLARGFF","created_at":"2026-05-18T12:30:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:DOLARGFFFCDOQKFZWZZZL4RRZP","target":"record","payload":{"canonical_record":{"source":{"id":"1601.05680","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-01-21T15:37:54Z","cross_cats_sorted":[],"title_canon_sha256":"0006d09ed6d64f8621e4d8d385bf5f55684f12963570c32a9377a4abba2c662e","abstract_canon_sha256":"8397be97370ffc8dc4ef50a17d3047743bd10c9d978150c72edd5779b363ec74"},"schema_version":"1.0"},"canonical_sha256":"1b960898a52886e828b9b67395f231cbc839b4b0126573c4fb5cf1be47f5d217","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:13.249436Z","signature_b64":"47Otl8lZhAV1HeXEoxAt9r+ieq7bXETsR/5AzQe43ZpMoVfXDcEuamN65R+Rt0qGzD9yC78U7/r4tfKkbNPJBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1b960898a52886e828b9b67395f231cbc839b4b0126573c4fb5cf1be47f5d217","last_reissued_at":"2026-05-18T01:22:13.249022Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:13.249022Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1601.05680","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":"2026-05-18T01:22:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MmDiWVVMtKbn48TcnZ6gDe1kJbVQPngMuSsW7i3MZZoHjVsg0LI2EHCkzuzBOqs+rZl7EnCBnDn6bOEY0iYvDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T11:23:41.063714Z"},"content_sha256":"ce902d8114cdd376521db1a4ae6242a3c5ffa0ab295ecc7dbb2a9f70c1c4d6e5","schema_version":"1.0","event_id":"sha256:ce902d8114cdd376521db1a4ae6242a3c5ffa0ab295ecc7dbb2a9f70c1c4d6e5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:DOLARGFFFCDOQKFZWZZZL4RRZP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Weak Galerkin Finite Element Scheme for solving the stationary Stokes Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Qilong Zhai, Ran Zhang, Ruishu Wang, Xiaoshen Wang","submitted_at":"2016-01-21T15:37:54Z","abstract_excerpt":"A weak Galerkin (WG) finite element method for solving the stationary Stokes equations in two- or three- dimensional spaces by using discontinuous piecewise polynomials is developed and analyzed. The variational form we considered is based on two gradient operators which is different from the usual gradient-divergence operators. The WG method is highly flexible by allowing the use of discontinuous functions on arbitrary polygons or polyhedra with certain shape regularity. Optimal-order error estimates are established for the corresponding WG finite element solutions in various norms. Numerical"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.05680","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":"2026-05-18T01:22:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"coPuVHqigCMNcmdBZfctv2XQosz1QuExwGuCQJ9uKFWD3Wfy5Tm4bbQ/9dKnnnCA7g1y1UXFqN7cGon3c4wzDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T11:23:41.064053Z"},"content_sha256":"d5f7de3ec50ae33749ecf3ac6645ba77063106653b8b0cc0f98d765c59a86e65","schema_version":"1.0","event_id":"sha256:d5f7de3ec50ae33749ecf3ac6645ba77063106653b8b0cc0f98d765c59a86e65"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DOLARGFFFCDOQKFZWZZZL4RRZP/bundle.json","state_url":"https://pith.science/pith/DOLARGFFFCDOQKFZWZZZL4RRZP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DOLARGFFFCDOQKFZWZZZL4RRZP/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":"2026-07-03T11:23:41Z","links":{"resolver":"https://pith.science/pith/DOLARGFFFCDOQKFZWZZZL4RRZP","bundle":"https://pith.science/pith/DOLARGFFFCDOQKFZWZZZL4RRZP/bundle.json","state":"https://pith.science/pith/DOLARGFFFCDOQKFZWZZZL4RRZP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DOLARGFFFCDOQKFZWZZZL4RRZP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:DOLARGFFFCDOQKFZWZZZL4RRZP","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":"8397be97370ffc8dc4ef50a17d3047743bd10c9d978150c72edd5779b363ec74","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-01-21T15:37:54Z","title_canon_sha256":"0006d09ed6d64f8621e4d8d385bf5f55684f12963570c32a9377a4abba2c662e"},"schema_version":"1.0","source":{"id":"1601.05680","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.05680","created_at":"2026-05-18T01:22:13Z"},{"alias_kind":"arxiv_version","alias_value":"1601.05680v1","created_at":"2026-05-18T01:22:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.05680","created_at":"2026-05-18T01:22:13Z"},{"alias_kind":"pith_short_12","alias_value":"DOLARGFFFCDO","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_16","alias_value":"DOLARGFFFCDOQKFZ","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_8","alias_value":"DOLARGFF","created_at":"2026-05-18T12:30:12Z"}],"graph_snapshots":[{"event_id":"sha256:d5f7de3ec50ae33749ecf3ac6645ba77063106653b8b0cc0f98d765c59a86e65","target":"graph","created_at":"2026-05-18T01:22:13Z","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":"A weak Galerkin (WG) finite element method for solving the stationary Stokes equations in two- or three- dimensional spaces by using discontinuous piecewise polynomials is developed and analyzed. The variational form we considered is based on two gradient operators which is different from the usual gradient-divergence operators. The WG method is highly flexible by allowing the use of discontinuous functions on arbitrary polygons or polyhedra with certain shape regularity. Optimal-order error estimates are established for the corresponding WG finite element solutions in various norms. Numerical","authors_text":"Qilong Zhai, Ran Zhang, Ruishu Wang, Xiaoshen Wang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-01-21T15:37:54Z","title":"A Weak Galerkin Finite Element Scheme for solving the stationary Stokes Equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.05680","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:ce902d8114cdd376521db1a4ae6242a3c5ffa0ab295ecc7dbb2a9f70c1c4d6e5","target":"record","created_at":"2026-05-18T01:22:13Z","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":"8397be97370ffc8dc4ef50a17d3047743bd10c9d978150c72edd5779b363ec74","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-01-21T15:37:54Z","title_canon_sha256":"0006d09ed6d64f8621e4d8d385bf5f55684f12963570c32a9377a4abba2c662e"},"schema_version":"1.0","source":{"id":"1601.05680","kind":"arxiv","version":1}},"canonical_sha256":"1b960898a52886e828b9b67395f231cbc839b4b0126573c4fb5cf1be47f5d217","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1b960898a52886e828b9b67395f231cbc839b4b0126573c4fb5cf1be47f5d217","first_computed_at":"2026-05-18T01:22:13.249022Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:22:13.249022Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"47Otl8lZhAV1HeXEoxAt9r+ieq7bXETsR/5AzQe43ZpMoVfXDcEuamN65R+Rt0qGzD9yC78U7/r4tfKkbNPJBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:22:13.249436Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.05680","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ce902d8114cdd376521db1a4ae6242a3c5ffa0ab295ecc7dbb2a9f70c1c4d6e5","sha256:d5f7de3ec50ae33749ecf3ac6645ba77063106653b8b0cc0f98d765c59a86e65"],"state_sha256":"a99c26f3800dc265c086c15c1d82b36d3fb5c8a08f1072ce4f61ece023429147"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yJ/76sn4L/zMP8xXXucgCce3DHeY3y1aBrbQOnHnVczaTCK1uGWtLnhokMS7lSX1+PP37ystEovheIf/GwcfDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-03T11:23:41.066032Z","bundle_sha256":"192da7cd78d7fd12da8e6f2d9eaf671ffd805f10d0d9d67fe7131925735282c2"}}