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pith:DDIJZPJU

pith:2025:DDIJZPJUWFRBO3RVW4B76KBAZO

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Local Multilevel Preconditioned Jacobi-Davidson Method for Elliptic Eigenvalue Problems on Adaptive Meshes

Jianing Guo, Qigang Liang, Xuejun Xu

A local multilevel preconditioned Jacobi-Davidson method achieves optimal O(N) complexity and uniform convergence for elliptic eigenvalue problems on adaptive meshes.

arxiv:2511.18996 v2 · 2025-11-24 · math.NA · cs.NA

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Claims

C1strongest 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.

C2weakest 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.

C3one 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.

References

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[1] 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 2008 · doi:10.1007/s00211-008-0169-3

[2] 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 2002 · doi:10.1137/s1064827501383713

[3] 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 1934 · doi:10.1093/imanum/dru059

[4] 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 2015 · doi:10.1007/s00211-014-0671-8

[5] 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 2020 · doi:10.1090/mcom/3549

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First computed	2026-05-18T03:09:32.987504Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

18d09cbd34b162176e35b703ff2820cbbc2ec2b562267fc9d40cabd6643b664d

Aliases

arxiv: 2511.18996 · arxiv_version: 2511.18996v2 · doi: 10.48550/arxiv.2511.18996 · pith_short_12: DDIJZPJUWFRB · pith_short_16: DDIJZPJUWFRBO3RV · pith_short_8: DDIJZPJU

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curl -sH 'Accept: application/ld+json' https://pith.science/pith/DDIJZPJUWFRBO3RVW4B76KBAZO \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 18d09cbd34b162176e35b703ff2820cbbc2ec2b562267fc9d40cabd6643b664d

Canonical record JSON

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    "cross_cats_sorted": [
      "cs.NA"
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    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NA",
    "submitted_at": "2025-11-24T11:21:20Z",
    "title_canon_sha256": "65f303106e46da335db43086ef0fb1a3c16f4fc9f2ec3eefbbd5a94a6a29442e"
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}