{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:T322CL2VD256FXV3AL75QBZZTX","short_pith_number":"pith:T322CL2V","canonical_record":{"source":{"id":"2603.14026","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2026-03-14T16:57:50Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"aab751915e86ab70c8ff1a58c8011f2bd7bd37b402f4112f57b2b54e27ffdcd8","abstract_canon_sha256":"42e22666d60d932d7cd392d86480727f3946b7c9f9f0cce86e72c0a2635c83f7"},"schema_version":"1.0"},"canonical_sha256":"9ef5a12f551ebbe2debb02ffd807399dd861447c2af2d4140ef9765eb5f00479","source":{"kind":"arxiv","id":"2603.14026","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2603.14026","created_at":"2026-05-29T01:05:06Z"},{"alias_kind":"arxiv_version","alias_value":"2603.14026v3","created_at":"2026-05-29T01:05:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2603.14026","created_at":"2026-05-29T01:05:06Z"},{"alias_kind":"pith_short_12","alias_value":"T322CL2VD256","created_at":"2026-05-29T01:05:06Z"},{"alias_kind":"pith_short_16","alias_value":"T322CL2VD256FXV3","created_at":"2026-05-29T01:05:06Z"},{"alias_kind":"pith_short_8","alias_value":"T322CL2V","created_at":"2026-05-29T01:05:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:T322CL2VD256FXV3AL75QBZZTX","target":"record","payload":{"canonical_record":{"source":{"id":"2603.14026","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2026-03-14T16:57:50Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"aab751915e86ab70c8ff1a58c8011f2bd7bd37b402f4112f57b2b54e27ffdcd8","abstract_canon_sha256":"42e22666d60d932d7cd392d86480727f3946b7c9f9f0cce86e72c0a2635c83f7"},"schema_version":"1.0"},"canonical_sha256":"9ef5a12f551ebbe2debb02ffd807399dd861447c2af2d4140ef9765eb5f00479","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-29T01:05:06.957552Z","signature_b64":"mqMrOlDE56obZFCW/Lty1Rsv28JOuyJ1Jd8Bqch6I2Eyg1KMyKj2RHuY4ITCEfnm/H7KA1jvrl0/O6hpnc1rBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9ef5a12f551ebbe2debb02ffd807399dd861447c2af2d4140ef9765eb5f00479","last_reissued_at":"2026-05-29T01:05:06.956563Z","signature_status":"signed_v1","first_computed_at":"2026-05-29T01:05:06.956563Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2603.14026","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-29T01:05:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZDZqSkHXuhGtubm8YBeywHcdg+dxraEASbzO17xhKKxKcY7RbdFd0m1QsCyyeKkX1QGZ05hbvob+lMX94fHwAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T04:49:00.816545Z"},"content_sha256":"a9e0f501e60ac9adaa1cf5f36ffc297555f4926073504f5dfc034478b7903ff5","schema_version":"1.0","event_id":"sha256:a9e0f501e60ac9adaa1cf5f36ffc297555f4926073504f5dfc034478b7903ff5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:T322CL2VD256FXV3AL75QBZZTX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Mixed Finite Element Method for the Dirichlet Vector Laplacian in Three Dimensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Mixed finite element method establishes well-posedness for the three-dimensional Dirichlet vector Laplacian via a non-standard vorticity space.","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Peiyang Yu, Ralf Hiptmair, Tianwei Yu","submitted_at":"2026-03-14T16:57:50Z","abstract_excerpt":"This work establishes the well-posedness and a priori error analysis for the mixed FEEC-type finite element approximation of the three-dimensional vector Laplace boundary value problem subject to the Dirichlet boundary condition. The Dirichlet condition disrupts the structure of the standard de Rham complex, requiring the vorticity to be sought in a non-standard function space to achieve well-posedness. We derive error estimates that confirm the numerically observed suboptimal convergence rates. In particular, by developing a discrete Caccioppoli-type inequality for discrete curl-harmonic func"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"This work establishes the well-posedness and a priori error analysis for the mixed FEEC-type finite element approximation of the three-dimensional vector Laplace boundary value problem subject to the Dirichlet boundary condition.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The discrete Caccioppoli-type inequality for discrete curl-harmonic functions holds in the chosen finite element spaces on general three-dimensional domains.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A mixed FEEC finite element approximation for the 3D Dirichlet vector Laplacian is well-posed, achieving (k-1/2)-order convergence in the energy norm on general domains and k-order convergence in L2 on convex domains for polynomial degree k.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Mixed finite element method establishes well-posedness for the three-dimensional Dirichlet vector Laplacian via a non-standard vorticity space.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"f278967d0a79d83073d3bcb1d9d03b07557e7bdb8773cef918198a6daf7773b1"},"source":{"id":"2603.14026","kind":"arxiv","version":3},"verdict":{"id":"39766368-eca0-422b-98dc-f501e7bae67e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T11:20:10.376437Z","strongest_claim":"This work establishes the well-posedness and a priori error analysis for the mixed FEEC-type finite element approximation of the three-dimensional vector Laplace boundary value problem subject to the Dirichlet boundary condition.","one_line_summary":"A mixed FEEC finite element approximation for the 3D Dirichlet vector Laplacian is well-posed, achieving (k-1/2)-order convergence in the energy norm on general domains and k-order convergence in L2 on convex domains for polynomial degree k.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The discrete Caccioppoli-type inequality for discrete curl-harmonic functions holds in the chosen finite element spaces on general three-dimensional domains.","pith_extraction_headline":"Mixed finite element method establishes well-posedness for the three-dimensional Dirichlet vector Laplacian via a non-standard vorticity space."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2603.14026/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"1879b34fd325632c8290bd7b15fff788c580c8d8b17202115044971c3c3644ae"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":"39766368-eca0-422b-98dc-f501e7bae67e"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-29T01:05:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4TbBklISUX+2Ea87p5LonkI4xg2bd/xuQ44YGjyPj5GyYm9x2ZZbeX2H+vsCT83mUUuwoZhfHMNA5SIlGQz4AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T04:49:00.817489Z"},"content_sha256":"ec8db95bf0d0222d81770a48c3593bb7b07b4e8bbfa9218adc58c084c3441638","schema_version":"1.0","event_id":"sha256:ec8db95bf0d0222d81770a48c3593bb7b07b4e8bbfa9218adc58c084c3441638"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/T322CL2VD256FXV3AL75QBZZTX/bundle.json","state_url":"https://pith.science/pith/T322CL2VD256FXV3AL75QBZZTX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/T322CL2VD256FXV3AL75QBZZTX/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":"2026-06-10T04:49:00Z","links":{"resolver":"https://pith.science/pith/T322CL2VD256FXV3AL75QBZZTX","bundle":"https://pith.science/pith/T322CL2VD256FXV3AL75QBZZTX/bundle.json","state":"https://pith.science/pith/T322CL2VD256FXV3AL75QBZZTX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/T322CL2VD256FXV3AL75QBZZTX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:T322CL2VD256FXV3AL75QBZZTX","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":"42e22666d60d932d7cd392d86480727f3946b7c9f9f0cce86e72c0a2635c83f7","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2026-03-14T16:57:50Z","title_canon_sha256":"aab751915e86ab70c8ff1a58c8011f2bd7bd37b402f4112f57b2b54e27ffdcd8"},"schema_version":"1.0","source":{"id":"2603.14026","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2603.14026","created_at":"2026-05-29T01:05:06Z"},{"alias_kind":"arxiv_version","alias_value":"2603.14026v3","created_at":"2026-05-29T01:05:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2603.14026","created_at":"2026-05-29T01:05:06Z"},{"alias_kind":"pith_short_12","alias_value":"T322CL2VD256","created_at":"2026-05-29T01:05:06Z"},{"alias_kind":"pith_short_16","alias_value":"T322CL2VD256FXV3","created_at":"2026-05-29T01:05:06Z"},{"alias_kind":"pith_short_8","alias_value":"T322CL2V","created_at":"2026-05-29T01:05:06Z"}],"graph_snapshots":[{"event_id":"sha256:ec8db95bf0d0222d81770a48c3593bb7b07b4e8bbfa9218adc58c084c3441638","target":"graph","created_at":"2026-05-29T01:05:06Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"This work establishes the well-posedness and a priori error analysis for the mixed FEEC-type finite element approximation of the three-dimensional vector Laplace boundary value problem subject to the Dirichlet boundary condition."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The discrete Caccioppoli-type inequality for discrete curl-harmonic functions holds in the chosen finite element spaces on general three-dimensional domains."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"A mixed FEEC finite element approximation for the 3D Dirichlet vector Laplacian is well-posed, achieving (k-1/2)-order convergence in the energy norm on general domains and k-order convergence in L2 on convex domains for polynomial degree k."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Mixed finite element method establishes well-posedness for the three-dimensional Dirichlet vector Laplacian via a non-standard vorticity space."}],"snapshot_sha256":"f278967d0a79d83073d3bcb1d9d03b07557e7bdb8773cef918198a6daf7773b1"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"1879b34fd325632c8290bd7b15fff788c580c8d8b17202115044971c3c3644ae"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2603.14026/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This work establishes the well-posedness and a priori error analysis for the mixed FEEC-type finite element approximation of the three-dimensional vector Laplace boundary value problem subject to the Dirichlet boundary condition. The Dirichlet condition disrupts the structure of the standard de Rham complex, requiring the vorticity to be sought in a non-standard function space to achieve well-posedness. We derive error estimates that confirm the numerically observed suboptimal convergence rates. In particular, by developing a discrete Caccioppoli-type inequality for discrete curl-harmonic func","authors_text":"Peiyang Yu, Ralf Hiptmair, Tianwei Yu","cross_cats":["cs.NA"],"headline":"Mixed finite element method establishes well-posedness for the three-dimensional Dirichlet vector Laplacian via a non-standard vorticity space.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2026-03-14T16:57:50Z","title":"A Mixed Finite Element Method for the Dirichlet Vector Laplacian in Three Dimensions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2603.14026","kind":"arxiv","version":3},"verdict":{"created_at":"2026-05-15T11:20:10.376437Z","id":"39766368-eca0-422b-98dc-f501e7bae67e","model_set":{"reader":"grok-4.3"},"one_line_summary":"A mixed FEEC finite element approximation for the 3D Dirichlet vector Laplacian is well-posed, achieving (k-1/2)-order convergence in the energy norm on general domains and k-order convergence in L2 on convex domains for polynomial degree k.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Mixed finite element method establishes well-posedness for the three-dimensional Dirichlet vector Laplacian via a non-standard vorticity space.","strongest_claim":"This work establishes the well-posedness and a priori error analysis for the mixed FEEC-type finite element approximation of the three-dimensional vector Laplace boundary value problem subject to the Dirichlet boundary condition.","weakest_assumption":"The discrete Caccioppoli-type inequality for discrete curl-harmonic functions holds in the chosen finite element spaces on general three-dimensional domains."}},"verdict_id":"39766368-eca0-422b-98dc-f501e7bae67e"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:a9e0f501e60ac9adaa1cf5f36ffc297555f4926073504f5dfc034478b7903ff5","target":"record","created_at":"2026-05-29T01:05:06Z","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":"42e22666d60d932d7cd392d86480727f3946b7c9f9f0cce86e72c0a2635c83f7","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2026-03-14T16:57:50Z","title_canon_sha256":"aab751915e86ab70c8ff1a58c8011f2bd7bd37b402f4112f57b2b54e27ffdcd8"},"schema_version":"1.0","source":{"id":"2603.14026","kind":"arxiv","version":3}},"canonical_sha256":"9ef5a12f551ebbe2debb02ffd807399dd861447c2af2d4140ef9765eb5f00479","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9ef5a12f551ebbe2debb02ffd807399dd861447c2af2d4140ef9765eb5f00479","first_computed_at":"2026-05-29T01:05:06.956563Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-29T01:05:06.956563Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mqMrOlDE56obZFCW/Lty1Rsv28JOuyJ1Jd8Bqch6I2Eyg1KMyKj2RHuY4ITCEfnm/H7KA1jvrl0/O6hpnc1rBw==","signature_status":"signed_v1","signed_at":"2026-05-29T01:05:06.957552Z","signed_message":"canonical_sha256_bytes"},"source_id":"2603.14026","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a9e0f501e60ac9adaa1cf5f36ffc297555f4926073504f5dfc034478b7903ff5","sha256:ec8db95bf0d0222d81770a48c3593bb7b07b4e8bbfa9218adc58c084c3441638"],"state_sha256":"4550fd423c9feaff09e03d9ca234dadbbd18efc33722a3de8bae8eacc0af73c1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vkljnV1WjRniWGCjYjLq4/UZFo7F9yARwjd4nfmJNaDVB3o8gIEwnmox3/92loRjFWQa22/cRFwbXA2TM4emCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T04:49:00.822200Z","bundle_sha256":"ce0748ecfd099baa23345ac0e2e488e2c99f602b2e23b9f2558a41e1d4210785"}}