{"paper":{"title":"BDF2-type integrator for Landau-Lifshitz-Gilbert equation in micromagnetics: a-priori error estimates","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"A linear BDF2 scheme paired with finite elements achieves optimal-order convergence for the Landau-Lifshitz-Gilbert equation.","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Dirk Praetorius, Michael Feischl, Michele Ald\\'e","submitted_at":"2026-05-06T16:56:38Z","abstract_excerpt":"We consider the Landau-Lifshitz-Gilbert equation (LLG), which models time-dependent micromagnetic phenomena. We analyze a fully discrete scheme that combines first-order finite elements in space with a BDF2 method in time. The method requires the solution of only one linear system of equations per time step and does not enforce the pointwise unit-length constraint of the magnetization. While unconditional weak convergence has been analyzed in an earlier work, we now prove optimal-order convergence rates under sufficient regularity assumptions on the exact solution and the external field. In co"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"In combination with our previous work, this establishes the first higher-order-in-time and linear integrator that converges both to weak and strong solutions of LLG.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"Sufficient regularity assumptions on the exact solution and the external field are required for the optimal-order a-priori error estimates to hold.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A linear BDF2 finite-element integrator for the LLG equation achieves first-order spatial and second-order temporal convergence rates and converges to both weak and strong solutions.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A linear BDF2 scheme paired with finite elements achieves optimal-order convergence for the Landau-Lifshitz-Gilbert equation.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"ea6927d81aefc45c65ddf31bbc120bc003b7a7512242a0e5eb8d7de5b3808493"},"source":{"id":"2605.05129","kind":"arxiv","version":2},"verdict":{"id":"0fe8a8cd-ec5e-46db-a672-562513c1d3fc","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T15:49:24.570215Z","strongest_claim":"In combination with our previous work, this establishes the first higher-order-in-time and linear integrator that converges both to weak and strong solutions of LLG.","one_line_summary":"A linear BDF2 finite-element integrator for the LLG equation achieves first-order spatial and second-order temporal convergence rates and converges to both weak and strong solutions.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"Sufficient regularity assumptions on the exact solution and the external field are required for the optimal-order a-priori error estimates to hold.","pith_extraction_headline":"A linear BDF2 scheme paired with finite elements achieves optimal-order convergence for the Landau-Lifshitz-Gilbert equation."},"integrity":{"clean":false,"summary":{"advisory":0,"critical":1,"by_detector":{"doi_compliance":{"total":1,"advisory":0,"critical":1,"informational":0}},"informational":0},"endpoint":"/pith/2605.05129/integrity.json","findings":[{"note":"DOI '10.1007/978-3-540-85054-' as printed in the bibliography is syntactically invalid and cannot resolve.","detector":"doi_compliance","severity":"critical","ref_index":10,"audited_at":"2026-05-19T13:49:15.581701Z","detected_doi":"10.1007/978-3-540-85054-","finding_type":"broken_identifier","verdict_class":"incontrovertible","detected_arxiv_id":null}],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T10:36:56.562986Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T21:31:19.510990Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T13:49:15.581701Z","status":"completed","version":"1.0.0","findings_count":1}],"snapshot_sha256":"b32ec979e5b9b80f829871589825cfc1141bc382dee1d2feee48e93df55bfaef"},"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"}