{"paper":{"title":"Local Multilevel Preconditioned Jacobi-Davidson Method for Elliptic Eigenvalue Problems on Adaptive Meshes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A local multilevel preconditioned Jacobi-Davidson method achieves optimal O(N) complexity and uniform convergence for elliptic eigenvalue problems on adaptive meshes.","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Jianing Guo, Qigang Liang, Xuejun Xu","submitted_at":"2025-11-24T11:21:20Z","abstract_excerpt":"In this work, we propose an efficient adaptive multilevel preconditioned Jacobi-Davidson (PJD) method for eigenvalue problems with singularity. Our multilevel method utilizes a local smoothing strategy to solve the preconditioned Jacobi-Davidson algebraic systems arising from adaptive finite element methods (AFEM). As a result, the algorithm holds optimal computational complexity $O(N)$. The theoretical analysis reveals that our method has a uniform convergence rate with respect to mesh levels and degrees of freedom. Further, the convergence rate is not affected by highly discontinuous coeffic"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Our multilevel method utilizes a local smoothing strategy to solve the preconditioned Jacobi-Davidson algebraic systems arising from adaptive finite element methods (AFEM). As a result, the algorithm holds optimal computational complexity O(N). The theoretical analysis reveals that our method has a uniform convergence rate with respect to mesh levels and degrees of freedom. Further, the convergence rate is not affected by highly discontinuous coefficients within the domain.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The local smoothing strategy on the newest mesh elements produces a sufficiently accurate preconditioner for the Jacobi-Davidson correction equation at every level; this is invoked in the complexity and convergence analysis but its precise error bound relative to the global residual is not visible in the abstract.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A local multilevel preconditioned Jacobi-Davidson solver for singular elliptic eigenvalue problems on adaptive meshes achieves O(N) complexity and uniform convergence independent of mesh level and coefficient discontinuities.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A local multilevel preconditioned Jacobi-Davidson method achieves optimal O(N) complexity and uniform convergence for elliptic eigenvalue problems on adaptive meshes.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"91113d7eaa20d149e3bc88b99440de14d82a4cc7b6b64524f6f454c0b5df0807"},"source":{"id":"2511.18996","kind":"arxiv","version":2},"verdict":{"id":"0b1412cb-7a73-4c6f-83fe-6ed42461c050","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-17T05:36:46.793909Z","strongest_claim":"Our multilevel method utilizes a local smoothing strategy to solve the preconditioned Jacobi-Davidson algebraic systems arising from adaptive finite element methods (AFEM). As a result, the algorithm holds optimal computational complexity O(N). The theoretical analysis reveals that our method has a uniform convergence rate with respect to mesh levels and degrees of freedom. Further, the convergence rate is not affected by highly discontinuous coefficients within the domain.","one_line_summary":"A local multilevel preconditioned Jacobi-Davidson solver for singular elliptic eigenvalue problems on adaptive meshes achieves O(N) complexity and uniform convergence independent of mesh level and coefficient discontinuities.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The local smoothing strategy on the newest mesh elements produces a sufficiently accurate preconditioner for the Jacobi-Davidson correction equation at every level; this is invoked in the complexity and convergence analysis but its precise error bound relative to the global residual is not visible in the abstract.","pith_extraction_headline":"A local multilevel preconditioned Jacobi-Davidson method achieves optimal O(N) complexity and uniform convergence for elliptic eigenvalue problems on adaptive meshes."},"references":{"count":21,"sample":[{"doi":"10.1007/s00211-008-0169-3","year":2008,"title":"Dai, X., Xu, J., Zhou, A.: Convergence and optimal complexity of adaptive finite element eigenvalue computations. Numer. Math.110(3), 313–355 (2008) https: //doi.org/10.1007/s00211-008-0169-3","work_id":"7699901b-9dc7-4ed1-92d0-1411d58d4a68","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1137/s1064827501383713","year":2002,"title":"Chen, Z., Dai, S.: On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients. SIAM J. Sci. Comput.24(2), 443–462 (2002) https://doi.org/10.1137/S106482","work_id":"2978eb4e-e755-4a5f-918a-0d0e6d17e1b2","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1093/imanum/dru059","year":1934,"title":"Dai, X., He, L., Zhou, A.: Convergence and quasi-optimal complexity of adaptive finite element computations for multiple eigenvalues. IMA J. Numer. Anal.35(4), 1934–1977 (2015) https://doi.org/10.1093","work_id":"b22e4aea-dc3c-432e-92a7-8fb38f8a2a68","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1007/s00211-014-0671-8","year":2015,"title":"Gallistl, D.: An optimal adaptive FEM for eigenvalue clusters. Numer. Math. 130(3), 467–496 (2015) https://doi.org/10.1007/s00211-014-0671-8","work_id":"d98d154a-0f68-4e51-b52f-1a6b680451ce","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1090/mcom/3549","year":2020,"title":"Canc` es, E., Dusson, G., Maday, Y., Stamm, B., Vohral´ ık, M.: Guaranteed a pos- teriori bounds for eigenvalues and eigenvectors: multiplicities and clusters. Math. Comp.89(326), 2563–2611 (2020) htt","work_id":"df14ebc8-27d0-496f-9f85-eff43219cb02","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":21,"snapshot_sha256":"db310a7ba09275c84ab2345a9a0c9934665bd1e5ded4719c2de48f73d0e2081b","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"520c3eee13541d3b133131a8dc73e3002fb3e8f4c5952a6535cb48a8badbd8cd"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}