{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:76GDA5GCDC2QB24JEL4RQBTXYJ","short_pith_number":"pith:76GDA5GC","canonical_record":{"source":{"id":"1408.3328","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2014-08-14T16:14:48Z","cross_cats_sorted":[],"title_canon_sha256":"d064603b5adce50a83b4f847271c5f7e9d8bf72f2d1c90b8dfbbb462ebcb1084","abstract_canon_sha256":"0f408d81296d16a29b0fbc1fe4abdeffbcef7ad61bab8ecfe00e00cb7f3b4f58"},"schema_version":"1.0"},"canonical_sha256":"ff8c3074c218b500eb8922f9180677c27eb2da6a5a1dec86a41b1a28ac39d356","source":{"kind":"arxiv","id":"1408.3328","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.3328","created_at":"2026-05-18T01:13:31Z"},{"alias_kind":"arxiv_version","alias_value":"1408.3328v2","created_at":"2026-05-18T01:13:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.3328","created_at":"2026-05-18T01:13:31Z"},{"alias_kind":"pith_short_12","alias_value":"76GDA5GCDC2Q","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_16","alias_value":"76GDA5GCDC2QB24J","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_8","alias_value":"76GDA5GC","created_at":"2026-05-18T12:28:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:76GDA5GCDC2QB24JEL4RQBTXYJ","target":"record","payload":{"canonical_record":{"source":{"id":"1408.3328","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2014-08-14T16:14:48Z","cross_cats_sorted":[],"title_canon_sha256":"d064603b5adce50a83b4f847271c5f7e9d8bf72f2d1c90b8dfbbb462ebcb1084","abstract_canon_sha256":"0f408d81296d16a29b0fbc1fe4abdeffbcef7ad61bab8ecfe00e00cb7f3b4f58"},"schema_version":"1.0"},"canonical_sha256":"ff8c3074c218b500eb8922f9180677c27eb2da6a5a1dec86a41b1a28ac39d356","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:13:31.154058Z","signature_b64":"G/In39AAbWoSP4pIlTqDK5/nJ2OVZ0pG3ZiWmDLCoWwkVeZkBS0ipNEFvpw4PXJ9+ZB/yebLFRh5EA2XznLMAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ff8c3074c218b500eb8922f9180677c27eb2da6a5a1dec86a41b1a28ac39d356","last_reissued_at":"2026-05-18T01:13:31.153390Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:13:31.153390Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1408.3328","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:13:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nArkYNBecH8sWTbnVVryrYsUWrqKBrmXvWf9R1cPUBJz+2c3aDxukWkM7lzI4fRAdreV8GI4uHG/0IkblxgMAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T12:33:02.011631Z"},"content_sha256":"32e654bf24e0af08836f4878ee5024596a4e06bdc1b00ebdb804912f07016668","schema_version":"1.0","event_id":"sha256:32e654bf24e0af08836f4878ee5024596a4e06bdc1b00ebdb804912f07016668"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:76GDA5GCDC2QB24JEL4RQBTXYJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Robust exponential convergence of $hp$-FEM in balanced norms for singularly perturbed reaction-diffusion equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Christos Xenophontos, Jens Markus Melenk","submitted_at":"2014-08-14T16:14:48Z","abstract_excerpt":"The $hp$-version of the finite element method is applied to a singularly perturbed reaction-diffusion equation posed in one- and two-dimensional domains with analytic boundary. On suitably designed \\emph{Spectral Boundary Layer meshes}, robust exponential convergence in a balanced norm is shown. This balanced norm is stronger than the energy norm in that the boundary layers are $O(1)$ uniformly in the singular perturbation parameter. Robust exponential convergence in the maximum norm is also established. The theoretical findings are illustrated with two numerical experiments."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.3328","kind":"arxiv","version":2},"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":"2026-05-18T01:13:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sM1iL8Ujmh2qFo9a2hMoicpklxuzuCCOE8ebdgSTLUaYsWjuEiaIwYGpbtL0L1Cw9qrtHDHngtjFqxRRXvakCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T12:33:02.011976Z"},"content_sha256":"b1cba5a402b7b531a23fb051d9950d7cbe9b43de10e4eb17bd4786ad9e96f21b","schema_version":"1.0","event_id":"sha256:b1cba5a402b7b531a23fb051d9950d7cbe9b43de10e4eb17bd4786ad9e96f21b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/76GDA5GCDC2QB24JEL4RQBTXYJ/bundle.json","state_url":"https://pith.science/pith/76GDA5GCDC2QB24JEL4RQBTXYJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/76GDA5GCDC2QB24JEL4RQBTXYJ/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":"2026-07-04T12:33:02Z","links":{"resolver":"https://pith.science/pith/76GDA5GCDC2QB24JEL4RQBTXYJ","bundle":"https://pith.science/pith/76GDA5GCDC2QB24JEL4RQBTXYJ/bundle.json","state":"https://pith.science/pith/76GDA5GCDC2QB24JEL4RQBTXYJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/76GDA5GCDC2QB24JEL4RQBTXYJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:76GDA5GCDC2QB24JEL4RQBTXYJ","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":"0f408d81296d16a29b0fbc1fe4abdeffbcef7ad61bab8ecfe00e00cb7f3b4f58","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2014-08-14T16:14:48Z","title_canon_sha256":"d064603b5adce50a83b4f847271c5f7e9d8bf72f2d1c90b8dfbbb462ebcb1084"},"schema_version":"1.0","source":{"id":"1408.3328","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.3328","created_at":"2026-05-18T01:13:31Z"},{"alias_kind":"arxiv_version","alias_value":"1408.3328v2","created_at":"2026-05-18T01:13:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.3328","created_at":"2026-05-18T01:13:31Z"},{"alias_kind":"pith_short_12","alias_value":"76GDA5GCDC2Q","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_16","alias_value":"76GDA5GCDC2QB24J","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_8","alias_value":"76GDA5GC","created_at":"2026-05-18T12:28:16Z"}],"graph_snapshots":[{"event_id":"sha256:b1cba5a402b7b531a23fb051d9950d7cbe9b43de10e4eb17bd4786ad9e96f21b","target":"graph","created_at":"2026-05-18T01:13:31Z","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":"The $hp$-version of the finite element method is applied to a singularly perturbed reaction-diffusion equation posed in one- and two-dimensional domains with analytic boundary. On suitably designed \\emph{Spectral Boundary Layer meshes}, robust exponential convergence in a balanced norm is shown. This balanced norm is stronger than the energy norm in that the boundary layers are $O(1)$ uniformly in the singular perturbation parameter. Robust exponential convergence in the maximum norm is also established. The theoretical findings are illustrated with two numerical experiments.","authors_text":"Christos Xenophontos, Jens Markus Melenk","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2014-08-14T16:14:48Z","title":"Robust exponential convergence of $hp$-FEM in balanced norms for singularly perturbed reaction-diffusion equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.3328","kind":"arxiv","version":2},"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":"sha256:32e654bf24e0af08836f4878ee5024596a4e06bdc1b00ebdb804912f07016668","target":"record","created_at":"2026-05-18T01:13:31Z","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":"0f408d81296d16a29b0fbc1fe4abdeffbcef7ad61bab8ecfe00e00cb7f3b4f58","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2014-08-14T16:14:48Z","title_canon_sha256":"d064603b5adce50a83b4f847271c5f7e9d8bf72f2d1c90b8dfbbb462ebcb1084"},"schema_version":"1.0","source":{"id":"1408.3328","kind":"arxiv","version":2}},"canonical_sha256":"ff8c3074c218b500eb8922f9180677c27eb2da6a5a1dec86a41b1a28ac39d356","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ff8c3074c218b500eb8922f9180677c27eb2da6a5a1dec86a41b1a28ac39d356","first_computed_at":"2026-05-18T01:13:31.153390Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:13:31.153390Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"G/In39AAbWoSP4pIlTqDK5/nJ2OVZ0pG3ZiWmDLCoWwkVeZkBS0ipNEFvpw4PXJ9+ZB/yebLFRh5EA2XznLMAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:13:31.154058Z","signed_message":"canonical_sha256_bytes"},"source_id":"1408.3328","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:32e654bf24e0af08836f4878ee5024596a4e06bdc1b00ebdb804912f07016668","sha256:b1cba5a402b7b531a23fb051d9950d7cbe9b43de10e4eb17bd4786ad9e96f21b"],"state_sha256":"b5d488a75153625d1527ae3f49b5e12254f2b9cd65eed539f6237ed6456f3e06"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aKwV1eKR4u6XSaOqHWq3JoYqLEkzv4xhqFXSaggrj6+l93ZI3S9JJFa5uSjWHn+AVWV3yF65nrGlPMfWWwo6AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-04T12:33:02.014042Z","bundle_sha256":"bf2f7e4b222c7e9919821a35d92ae9b26c7bf52f4d74409202aa22b7b19bc1b8"}}