{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:PNQ33NRUBNUVKOPOZK3LMWFSIL","short_pith_number":"pith:PNQ33NRU","canonical_record":{"source":{"id":"1811.02988","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2018-11-06T14:31:51Z","cross_cats_sorted":[],"title_canon_sha256":"a01ef457d027117c70640018bd89e91886b977cb77b24ae7780179d8f3ecb4ca","abstract_canon_sha256":"875ec849aac8e2ce3fecfdfcde821ee93161f0356ed49089b17796abbcbe96d0"},"schema_version":"1.0"},"canonical_sha256":"7b61bdb6340b695539eecab6b658b242d6b052e1d6fd196eb78eb30cc3b2c9e2","source":{"kind":"arxiv","id":"1811.02988","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.02988","created_at":"2026-05-18T00:01:20Z"},{"alias_kind":"arxiv_version","alias_value":"1811.02988v1","created_at":"2026-05-18T00:01:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.02988","created_at":"2026-05-18T00:01:20Z"},{"alias_kind":"pith_short_12","alias_value":"PNQ33NRUBNUV","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_16","alias_value":"PNQ33NRUBNUVKOPO","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_8","alias_value":"PNQ33NRU","created_at":"2026-05-18T12:32:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:PNQ33NRUBNUVKOPOZK3LMWFSIL","target":"record","payload":{"canonical_record":{"source":{"id":"1811.02988","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2018-11-06T14:31:51Z","cross_cats_sorted":[],"title_canon_sha256":"a01ef457d027117c70640018bd89e91886b977cb77b24ae7780179d8f3ecb4ca","abstract_canon_sha256":"875ec849aac8e2ce3fecfdfcde821ee93161f0356ed49089b17796abbcbe96d0"},"schema_version":"1.0"},"canonical_sha256":"7b61bdb6340b695539eecab6b658b242d6b052e1d6fd196eb78eb30cc3b2c9e2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:01:20.191607Z","signature_b64":"8j1XaaOQC/2m4QqDYQSJf1DXFMHGpflILqiHHbkGy5Mo8MwO6vDLUjbNR5iS+73HckESKu0sERfS5/w+EydmDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7b61bdb6340b695539eecab6b658b242d6b052e1d6fd196eb78eb30cc3b2c9e2","last_reissued_at":"2026-05-18T00:01:20.191161Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:01:20.191161Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1811.02988","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":"2026-05-18T00:01:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SEXSXoeW9lqhMoJ7H6abPasv82XdJ0sBJDoeVX0B0IgXKzh/V4yoM9Uu36bc/2dKmDJtwQ7Wv5/UYBB37QSFAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T03:18:44.973394Z"},"content_sha256":"2c26a0158500f71512a7d3c5e3e4e7fc24d0a6fa024371236ca931667b485f89","schema_version":"1.0","event_id":"sha256:2c26a0158500f71512a7d3c5e3e4e7fc24d0a6fa024371236ca931667b485f89"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:PNQ33NRUBNUVKOPOZK3LMWFSIL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Geometric multigrid methods for Darcy-Forchheimer flow in fractured porous media","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Andr\\'es Arrar\\'as, Carmen Rodrigo, Francisco J. Gaspar, Laura Portero","submitted_at":"2018-11-06T14:31:51Z","abstract_excerpt":"In this paper, we present a monolithic multigrid method for the efficient solution of flow problems in fractured porous media. Specifically, we consider a mixed-dimensional model which couples Darcy flow in the porous matrix with Forchheimer flow within the fractures. A suitable finite volume discretization permits to reduce the coupled problem to a system of nonlinear equations with a saddle point structure. In order to solve this system, we propose a full approximation scheme (FAS) multigrid solver that appropriately deals with the mixed-dimensional nature of the problem by using mixed-dimen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.02988","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":"2026-05-18T00:01:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yki3mMbfVjyqXySi44Xhtb9ll7Jef6H1HTOgCnIQ5n1e+aWCdhFEzJ4np7OCTWDMnLEoKleS2jVL1Tl/XvFJCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T03:18:44.974065Z"},"content_sha256":"da041f52a03fb6db1e71f737147b1ba1da7c056606efd20cc36acd16da3a06e0","schema_version":"1.0","event_id":"sha256:da041f52a03fb6db1e71f737147b1ba1da7c056606efd20cc36acd16da3a06e0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PNQ33NRUBNUVKOPOZK3LMWFSIL/bundle.json","state_url":"https://pith.science/pith/PNQ33NRUBNUVKOPOZK3LMWFSIL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PNQ33NRUBNUVKOPOZK3LMWFSIL/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":"2026-05-27T03:18:44Z","links":{"resolver":"https://pith.science/pith/PNQ33NRUBNUVKOPOZK3LMWFSIL","bundle":"https://pith.science/pith/PNQ33NRUBNUVKOPOZK3LMWFSIL/bundle.json","state":"https://pith.science/pith/PNQ33NRUBNUVKOPOZK3LMWFSIL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PNQ33NRUBNUVKOPOZK3LMWFSIL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:PNQ33NRUBNUVKOPOZK3LMWFSIL","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":"875ec849aac8e2ce3fecfdfcde821ee93161f0356ed49089b17796abbcbe96d0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2018-11-06T14:31:51Z","title_canon_sha256":"a01ef457d027117c70640018bd89e91886b977cb77b24ae7780179d8f3ecb4ca"},"schema_version":"1.0","source":{"id":"1811.02988","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.02988","created_at":"2026-05-18T00:01:20Z"},{"alias_kind":"arxiv_version","alias_value":"1811.02988v1","created_at":"2026-05-18T00:01:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.02988","created_at":"2026-05-18T00:01:20Z"},{"alias_kind":"pith_short_12","alias_value":"PNQ33NRUBNUV","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_16","alias_value":"PNQ33NRUBNUVKOPO","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_8","alias_value":"PNQ33NRU","created_at":"2026-05-18T12:32:46Z"}],"graph_snapshots":[{"event_id":"sha256:da041f52a03fb6db1e71f737147b1ba1da7c056606efd20cc36acd16da3a06e0","target":"graph","created_at":"2026-05-18T00:01:20Z","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 monolithic multigrid method for the efficient solution of flow problems in fractured porous media. Specifically, we consider a mixed-dimensional model which couples Darcy flow in the porous matrix with Forchheimer flow within the fractures. A suitable finite volume discretization permits to reduce the coupled problem to a system of nonlinear equations with a saddle point structure. In order to solve this system, we propose a full approximation scheme (FAS) multigrid solver that appropriately deals with the mixed-dimensional nature of the problem by using mixed-dimen","authors_text":"Andr\\'es Arrar\\'as, Carmen Rodrigo, Francisco J. Gaspar, Laura Portero","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2018-11-06T14:31:51Z","title":"Geometric multigrid methods for Darcy-Forchheimer flow in fractured porous media"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.02988","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":"sha256:2c26a0158500f71512a7d3c5e3e4e7fc24d0a6fa024371236ca931667b485f89","target":"record","created_at":"2026-05-18T00:01:20Z","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":"875ec849aac8e2ce3fecfdfcde821ee93161f0356ed49089b17796abbcbe96d0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2018-11-06T14:31:51Z","title_canon_sha256":"a01ef457d027117c70640018bd89e91886b977cb77b24ae7780179d8f3ecb4ca"},"schema_version":"1.0","source":{"id":"1811.02988","kind":"arxiv","version":1}},"canonical_sha256":"7b61bdb6340b695539eecab6b658b242d6b052e1d6fd196eb78eb30cc3b2c9e2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7b61bdb6340b695539eecab6b658b242d6b052e1d6fd196eb78eb30cc3b2c9e2","first_computed_at":"2026-05-18T00:01:20.191161Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:01:20.191161Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8j1XaaOQC/2m4QqDYQSJf1DXFMHGpflILqiHHbkGy5Mo8MwO6vDLUjbNR5iS+73HckESKu0sERfS5/w+EydmDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:01:20.191607Z","signed_message":"canonical_sha256_bytes"},"source_id":"1811.02988","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2c26a0158500f71512a7d3c5e3e4e7fc24d0a6fa024371236ca931667b485f89","sha256:da041f52a03fb6db1e71f737147b1ba1da7c056606efd20cc36acd16da3a06e0"],"state_sha256":"4cff7f9f08acf88d3558f276b2ef0ab616514b65a3a2154220df89e0dbdd657e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ahviWkiaZX95Kx8N0qD+iZwMaQgdxcsNdw20iykoPNDUpLn5xX3vE52H8/mSpMNjeecBlV0hwmfoqc3e/oswCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T03:18:44.977840Z","bundle_sha256":"6a8f4c626378f242cc0700931d6081bd5e9d86e307b1c8af790bd0558804e4b7"}}