{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:4PVZ7VNXWWZM7WPMZVSHB24P74","short_pith_number":"pith:4PVZ7VNX","canonical_record":{"source":{"id":"1608.03818","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-08-12T15:08:25Z","cross_cats_sorted":[],"title_canon_sha256":"3290678e69b96ca61e39a4d32f5c2bd638bd531415f0baa551b39563b7ccb492","abstract_canon_sha256":"b588e9d6aa7ffcfe43febe4a5a1cd34c875b6fae2b6af7f93e1af645571248bc"},"schema_version":"1.0"},"canonical_sha256":"e3eb9fd5b7b5b2cfd9eccd6470eb8fff2aff65ca0cd09c51ffe894ef5eb74c0e","source":{"kind":"arxiv","id":"1608.03818","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.03818","created_at":"2026-05-17T23:45:15Z"},{"alias_kind":"arxiv_version","alias_value":"1608.03818v1","created_at":"2026-05-17T23:45:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.03818","created_at":"2026-05-17T23:45:15Z"},{"alias_kind":"pith_short_12","alias_value":"4PVZ7VNXWWZM","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_16","alias_value":"4PVZ7VNXWWZM7WPM","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_8","alias_value":"4PVZ7VNX","created_at":"2026-05-18T12:29:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:4PVZ7VNXWWZM7WPMZVSHB24P74","target":"record","payload":{"canonical_record":{"source":{"id":"1608.03818","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-08-12T15:08:25Z","cross_cats_sorted":[],"title_canon_sha256":"3290678e69b96ca61e39a4d32f5c2bd638bd531415f0baa551b39563b7ccb492","abstract_canon_sha256":"b588e9d6aa7ffcfe43febe4a5a1cd34c875b6fae2b6af7f93e1af645571248bc"},"schema_version":"1.0"},"canonical_sha256":"e3eb9fd5b7b5b2cfd9eccd6470eb8fff2aff65ca0cd09c51ffe894ef5eb74c0e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:45:15.827358Z","signature_b64":"K0TrXHTK0iVoiuKZ7r+q5QsX85YiU7nJTuP9JXSq80T5ncEYvtkt3opbLa18ImxziVr/gtHHarP88yKuJa8GDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e3eb9fd5b7b5b2cfd9eccd6470eb8fff2aff65ca0cd09c51ffe894ef5eb74c0e","last_reissued_at":"2026-05-17T23:45:15.826836Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:45:15.826836Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1608.03818","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":"2026-05-17T23:45:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"24jockYuDvWsNaVKA6yQpVAHWQH94+y41qZZOs/OoNGk8ZVC2CEgqxU29/XDy95+i9XTyH1L/9HP1cRumwRHCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T12:16:39.606927Z"},"content_sha256":"748ab05bf5071f8e116a8df131a8ac2aa59f5aac2a329072d566f59e040f4ac5","schema_version":"1.0","event_id":"sha256:748ab05bf5071f8e116a8df131a8ac2aa59f5aac2a329072d566f59e040f4ac5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:4PVZ7VNXWWZM7WPMZVSHB24P74","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Super-convergence and post-processing for mixed finite element approximations of the wave equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Bogdan Radu, Herbert Egger","submitted_at":"2016-08-12T15:08:25Z","abstract_excerpt":"We consider the numerical approximation of acoustic wave propagation problems by mixed BDM(k+1)-P(k) finite elements on unstructured meshes. Optimal convergence of the discrete velocity and super-convergence of the pressure by one order are established. Based on these results, we propose a post-processing strategy that allows us to construct an improved pressure approximation from the numerical solution. Corresponding results are well-known for mixed finite element approximations of elliptic problems and we extend these analyses here to the hyperbolic problem under consideration. We also consi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.03818","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":"2026-05-17T23:45:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yJ0wqM3+ReHYbIGGawIgvK5tPyNAv/DrK8OvPNnDdj8aN9EkK7UMMi7r/KPhkw8kTby4/TZcFJ918pqZxluxDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T12:16:39.607291Z"},"content_sha256":"d049c15a9e049181e294d7871e9984579250b05ce69c40c590b753956cedbe28","schema_version":"1.0","event_id":"sha256:d049c15a9e049181e294d7871e9984579250b05ce69c40c590b753956cedbe28"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4PVZ7VNXWWZM7WPMZVSHB24P74/bundle.json","state_url":"https://pith.science/pith/4PVZ7VNXWWZM7WPMZVSHB24P74/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4PVZ7VNXWWZM7WPMZVSHB24P74/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":"2026-06-20T12:16:39Z","links":{"resolver":"https://pith.science/pith/4PVZ7VNXWWZM7WPMZVSHB24P74","bundle":"https://pith.science/pith/4PVZ7VNXWWZM7WPMZVSHB24P74/bundle.json","state":"https://pith.science/pith/4PVZ7VNXWWZM7WPMZVSHB24P74/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4PVZ7VNXWWZM7WPMZVSHB24P74/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:4PVZ7VNXWWZM7WPMZVSHB24P74","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":"b588e9d6aa7ffcfe43febe4a5a1cd34c875b6fae2b6af7f93e1af645571248bc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-08-12T15:08:25Z","title_canon_sha256":"3290678e69b96ca61e39a4d32f5c2bd638bd531415f0baa551b39563b7ccb492"},"schema_version":"1.0","source":{"id":"1608.03818","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.03818","created_at":"2026-05-17T23:45:15Z"},{"alias_kind":"arxiv_version","alias_value":"1608.03818v1","created_at":"2026-05-17T23:45:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.03818","created_at":"2026-05-17T23:45:15Z"},{"alias_kind":"pith_short_12","alias_value":"4PVZ7VNXWWZM","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_16","alias_value":"4PVZ7VNXWWZM7WPM","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_8","alias_value":"4PVZ7VNX","created_at":"2026-05-18T12:29:58Z"}],"graph_snapshots":[{"event_id":"sha256:d049c15a9e049181e294d7871e9984579250b05ce69c40c590b753956cedbe28","target":"graph","created_at":"2026-05-17T23:45:15Z","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":"We consider the numerical approximation of acoustic wave propagation problems by mixed BDM(k+1)-P(k) finite elements on unstructured meshes. Optimal convergence of the discrete velocity and super-convergence of the pressure by one order are established. Based on these results, we propose a post-processing strategy that allows us to construct an improved pressure approximation from the numerical solution. Corresponding results are well-known for mixed finite element approximations of elliptic problems and we extend these analyses here to the hyperbolic problem under consideration. We also consi","authors_text":"Bogdan Radu, Herbert Egger","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-08-12T15:08:25Z","title":"Super-convergence and post-processing for mixed finite element approximations of the wave equation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.03818","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":"sha256:748ab05bf5071f8e116a8df131a8ac2aa59f5aac2a329072d566f59e040f4ac5","target":"record","created_at":"2026-05-17T23:45:15Z","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":"b588e9d6aa7ffcfe43febe4a5a1cd34c875b6fae2b6af7f93e1af645571248bc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-08-12T15:08:25Z","title_canon_sha256":"3290678e69b96ca61e39a4d32f5c2bd638bd531415f0baa551b39563b7ccb492"},"schema_version":"1.0","source":{"id":"1608.03818","kind":"arxiv","version":1}},"canonical_sha256":"e3eb9fd5b7b5b2cfd9eccd6470eb8fff2aff65ca0cd09c51ffe894ef5eb74c0e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e3eb9fd5b7b5b2cfd9eccd6470eb8fff2aff65ca0cd09c51ffe894ef5eb74c0e","first_computed_at":"2026-05-17T23:45:15.826836Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:45:15.826836Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"K0TrXHTK0iVoiuKZ7r+q5QsX85YiU7nJTuP9JXSq80T5ncEYvtkt3opbLa18ImxziVr/gtHHarP88yKuJa8GDw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:45:15.827358Z","signed_message":"canonical_sha256_bytes"},"source_id":"1608.03818","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:748ab05bf5071f8e116a8df131a8ac2aa59f5aac2a329072d566f59e040f4ac5","sha256:d049c15a9e049181e294d7871e9984579250b05ce69c40c590b753956cedbe28"],"state_sha256":"b8d9e78e405c0dd69b668e024218d0b09c726b4933748712b408623a6ff5f231"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ipiio1piFGHaDNnVMKK7N/wrBfUerQkfzhGjgcRdLC9Swfm+HaXtjeHQCgZnRNVtD+q320qWhZgEFTahDVz/Bw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T12:16:39.609231Z","bundle_sha256":"d6e6f15dbea9b4f806e08a440e72ba8ba9f13fb4fe6577c906e9843aa4ec0f94"}}