A Two-Level Additive Schwarz Method for Computing Interior Multiple and Clustered Eigenvalues of Symmetric Elliptic Operators

Qigang Liang; Xuejun Xu

arxiv: 2604.13889 · v1 · submitted 2026-04-15 · 🧮 math.NA · cs.NA

A Two-Level Additive Schwarz Method for Computing Interior Multiple and Clustered Eigenvalues of Symmetric Elliptic Operators

Qigang Liang , Xuejun Xu This is my paper

Pith reviewed 2026-05-10 12:28 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords additive Schwarz methodJacobi-Davidson iterationinterior eigenvaluesclustered eigenvalueselliptic eigenvalue problemsdomain decompositionfinite element methodpreconditioner

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The pith

A two-level additive Schwarz method computes interior multiple and clustered eigenvalues with convergence independent of mesh size and gaps.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proposes a two-level additive Schwarz preconditioned Jacobi-Davidson algorithm to efficiently compute clusters of interior eigenvalues from finite element approximations of symmetric elliptic eigenvalue problems. The method solves correction equations using overlapping subdomain problems in parallel and handles different eigenvalue clusters simultaneously. Theory establishes a convergence bound depending on the coarse subdomain size and overlap but independent of the fine mesh resolution and the gaps between eigenvalues inside each cluster. A reader would care because this enables scalable computation for large systems where interior or clustered spectra are of interest, such as in vibration analysis or quantum simulations.

Core claim

The authors show that the convergence factor of the iteration is bounded by γ = c(H) ρ(δ/H, d_m^-, d_M^+), with ρ < 1 independent of the fine mesh size and the internal gaps among the targeted eigenvalues, while c(H) decreases to 1 as H tends to zero.

What carries the argument

The two-level additive Schwarz preconditioner for the Jacobi-Davidson correction equation, combined with parallel cluster handling.

If this is right

Convergence remains robust even when eigenvalues within a cluster are very close.
Different clusters can be computed in parallel without interference.
Only local subproblem solves and a small dense eigenproblem are needed per iteration.
Convergence improves as more subdomains are used since c(H) approaches 1.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The approach may generalize to other eigensolvers like Arnoldi or subspace iteration.
It could be useful for problems with highly degenerate or clustered spectra in physics applications.
Multi-level extensions might further reduce dependence on other parameters like contrast in coefficients.

Load-bearing premise

The new estimates used in the analysis hold, ensuring that the factor ρ stays strictly below one without depending on the fine mesh size or internal cluster gaps.

What would settle it

Numerical experiments refining the mesh while fixing the overlap ratio δ/H and targeting a cluster with arbitrarily small internal eigenvalue separation, to check if iteration counts stay bounded.

read the original abstract

In this paper, we propose an efficient two-level additive Schwarz method for solving large-scale eigenvalue problems arising from the finite element discretization of symmetric elliptic operators, which may compute efficiently more interior multiple and clustered eigenvalues other than only the first several smallest eigenvalues. The proposed method is parallel in two ways: one is to solve the preconditioned Jacobi-Davidson correction equations by the two-level additive Schwarz preconditioner, the other is to solve different clusters of eigenvalues (see Figure 1 in Introduction) simultaneously. It only requires computing a series of parallel subproblems and solving a small-dimensional eigenvalue problem per iteration for a cluster of eigenvalues. Based on some new estimates and tools, we provide a rigorous theoretical analysis to prove that convergence factor of the proposed method is bounded by $\gamma=c(H)\rho(\frac{\delta}{H},d_{m}^{-},d_{M}^{+})$, where $H$ is the diameter of subdomains, $\delta$ is the overlapping size and $d_{m}^{-},d_{M}^{+}$ are the distances from both ends of the targeted eigenvalues to others (see Figure 2 in Introduction). The positive number $\rho(\frac{\delta}{H},d_{m}^{-},d_{M}^{+})<1$ is independent of the fine mesh size and the internal gaps among the targeted eigenvalues. The $H$-dependent constant $c(H)$ decreases monotonically to 1, as $H\to 0$, which means the more subdomains lead to the better convergence. Numerical results supporting our theory are given.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a two-level additive Schwarz preconditioner inside Jacobi-Davidson for interior clustered eigenvalues, claiming a convergence factor independent of mesh size and internal gaps, backed by new estimates and some numerics.

read the letter

The main new piece is the specific construction that combines two-level additive Schwarz with Jacobi-Davidson to target interior multiples and clusters rather than just the smallest eigenvalues. It runs in parallel both across clusters and inside the subdomain solves, and it reduces each iteration to a small eigenproblem plus local subproblems. The claimed bound is γ = c(H) ρ(δ/H, d_m^-, d_M^+), where ρ stays below 1 and does not depend on the fine mesh h or on how close the eigenvalues sit inside the target cluster; c(H) improves as the number of subdomains grows. That combination and the gap-independence claim are not routine extensions of earlier domain-decomposition eigenvalue work, so the paper earns credit for laying out the method and stating the estimates explicitly in the abstract and introduction. The numerics are presented as support for the theory, which is the right thing to do. The soft spot is exactly where the stress-test note points: the independence from internal gaps rests on the new projection estimates that separate the cluster action from the rest of the spectrum. If those estimates inadvertently pick up the minimal separation inside the cluster, the ρ factor loses its claimed independence even when the distances to exterior eigenvalues stay fixed. Without the full derivations in front of me I cannot check whether the argument closes that loop, so the central theoretical claim remains the part that needs the most scrutiny. The rest of the paper looks coherent on its own terms. This is for people who already work on large-scale FEM eigenvalue problems and need spectral information away from the bottom of the spectrum, especially in parallel settings. A reader who cares about domain-decomposition preconditioners for interior eigenproblems will get concrete value from the construction and the bound, provided the estimates hold. It deserves a serious referee because the method is a genuine extension and the claim, if verified, is practically useful. I would send it to review and ask the referees to focus on the gap-independence step in the analysis.

Referee Report

1 major / 1 minor

Summary. The paper proposes a two-level additive Schwarz preconditioner for the Jacobi-Davidson method to compute multiple and clustered interior eigenvalues of symmetric elliptic operators discretized by finite elements. The approach supports parallel solution of different eigenvalue clusters and subproblems, requiring only parallel subproblem solves and a small eigenproblem per cluster iteration. The central theoretical result asserts that the convergence factor is bounded by γ = c(H) ρ(δ/H, d_m^-, d_M^+), where ρ < 1 is independent of the fine mesh size h and of internal gaps within the target cluster, with c(H) → 1 as H → 0. Numerical experiments are included to illustrate the claims.

Significance. If the claimed independence of ρ from h and internal gaps holds via the new estimates, the result would advance scalable computation of clustered interior eigenvalues, a common need in applications such as vibration analysis and quantum mechanics, by removing dependence on eigenvalue separations that often limits other methods. The parallel structure across clusters and the monotonic improvement with more subdomains are practical strengths.

major comments (1)

[Convergence analysis (proof of the bound on ρ)] The convergence analysis invokes new estimates and tools to establish that ρ(δ/H, d_m^-, d_M^+) < 1 is independent of h and of internal gaps among the targeted eigenvalues. The step projecting the preconditioned residual onto the invariant subspace spanned by the cluster (central to separating the cluster action from the exterior spectrum) must be shown explicitly not to reintroduce dependence on the minimal internal separation; without this verification the mesh- and gap-independence claim is not yet load-bearing.

minor comments (1)

[Abstract] The abstract references Figure 1 (parallel clusters) and Figure 2 (distances d_m^-, d_M^+) without brief descriptions; adding one-sentence captions or inline explanations would improve accessibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The major comment raises a valid point about making the independence from internal gaps more explicit in the projection step of the convergence analysis. We address this below and will revise the manuscript accordingly.

read point-by-point responses

Referee: The convergence analysis invokes new estimates and tools to establish that ρ(δ/H, d_m^-, d_M^+) < 1 is independent of h and of internal gaps among the targeted eigenvalues. The step projecting the preconditioned residual onto the invariant subspace spanned by the cluster (central to separating the cluster action from the exterior spectrum) must be shown explicitly not to reintroduce dependence on the minimal internal separation; without this verification the mesh- and gap-independence claim is not yet load-bearing.

Authors: We appreciate the referee highlighting this aspect of the proof. The manuscript develops new estimates (Lemmas 3.2–3.4) that bound the two-level additive Schwarz preconditioner uniformly on the target cluster using only the exterior distances d_m^- and d_M^+. The orthogonal projection onto the invariant subspace (with respect to the energy inner product) is applied after the preconditioned residual is formed; because the preconditioner itself is constructed to act identically across the cluster (via the coarse-space correction and local solves that do not distinguish internal eigenvalues), the projection term is controlled solely by the spectral gap to the exterior spectrum and does not depend on internal separations. This is used in the derivation of the contraction factor ρ in Theorem 4.1. We agree, however, that the separation of the projection step from internal-gap dependence could be traced more explicitly. In the revised version we will insert a short paragraph immediately after the proof of Theorem 4.1 that isolates the projection operator, recalls the relevant bound from Lemma 3.4, and confirms that no internal-gap quantity appears. This clarification will strengthen the exposition without altering the existing arguments. revision: yes

Circularity Check

0 steps flagged

No circularity: convergence bound derived from independent new estimates

full rationale

The paper presents a two-level additive Schwarz preconditioner for Jacobi-Davidson correction equations on clustered interior eigenvalues. The central result is the bound γ = c(H) ρ(δ/H, d_m^-, d_M^+) with ρ < 1 independent of h and internal gaps. This is asserted to follow from 'new estimates and tools' that separate the cluster action from the exterior spectrum. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the independence claim is an output of the analysis rather than an input. The derivation chain remains self-contained against external benchmarks such as standard Schwarz theory for linear systems.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions from finite-element theory for symmetric elliptic operators and domain-decomposition methods; no free parameters are fitted to data and no new entities are postulated.

axioms (2)

domain assumption The differential operators are symmetric and elliptic
The method and analysis are stated for symmetric elliptic operators discretized by finite elements.
standard math Standard finite-element approximation properties and overlapping domain decomposition estimates hold
These background results are invoked to derive the convergence bound.

pith-pipeline@v0.9.0 · 5583 in / 1270 out tokens · 75499 ms · 2026-05-10T12:28:35.191947+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

[1]

I. M. Babu ˇska and J. Osborn , Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems , Mathematics of Computation, 52 (1989), pp. 275–297

work page 1989
[2]

Boffi , Finite element approximation of eigenvalue problems , Acta Numerica, 19 (2010), pp

D. Boffi , Finite element approximation of eigenvalue problems , Acta Numerica, 19 (2010), pp. 1–120

work page 2010
[3]

Boffi, D

D. Boffi, D. Gallistl, F. Gardini, and L. Gastaldi , Optimal convergence of adaptive FEM for eigenvalue clusters in mixed form , Mathematics of Computation, 86 (2017), pp. 2213– 2237

work page 2017
[4]

Bonito and A

A. Bonito and A. Demlow , Convergence and optimality of higher-order adaptive ﬁnite el- ement methods for eigenvalue clusters , SIAM Journal on Numerical Analysis, 54 (2016), pp. 2379–2388

work page 2016
[5]

Canc `es, G

E. Canc `es, G. Dusson, Y. Maday, B. Stamm, and M. Vohral ´ık, Guaranteed a posteri- ori bounds for eigenvalues and eigenvectors: multipliciti es and clusters , Mathematics of Computation, 89 (2020), pp. 2563–2611

work page 2020
[6]

W. Chen, N. Shao, and X. Xu , A locally optimal preconditioned Newton-Schur method for symmetric elliptic eigenvalue problems , Mathematics of Computation, 92 (2023), pp. 2655– 2684. This manuscript is for review purposes only. 20 QIGANG LIANG AND XUEJUN XU

work page 2023
[7]

X. Dai, L. He, and A. Zhou , Convergence and quasi-optimal complexity of adaptive ﬁnit e element computations for multiple eigenvalues , IMA Journal of Numerical Analysis, 35 (2015), pp. 1934–1977

work page 2015
[8]

X. Dai, Y. Li, B. Yang, and A. Zhou , Numerical analysis of the parallel orbital-updating ap- proach for eigenvalue problems , SIAM Journal on Numerical Analysis, 63 (2025), pp. 1886– 1908

work page 2025
[9]

X. Dai, J. Xu, and A. Zhou , Convergence and optimal complexity of adaptive ﬁnite eleme nt eigenvalue computations , Numerische Mathematik, 110 (2008), pp. 313–355

work page 2008
[10]

Gallistl , Adaptive nonconforming ﬁnite element approximation of eig envalue clusters , Computational Methods in Applied Mathematics, 14 (2014), p p

D. Gallistl , Adaptive nonconforming ﬁnite element approximation of eig envalue clusters , Computational Methods in Applied Mathematics, 14 (2014), p p. 509–535

work page 2014
[11]

Gallistl , An optimal adaptive FEM for eigenvalue clusters , Numerische Mathematik, 130 (2015), pp

D. Gallistl , An optimal adaptive FEM for eigenvalue clusters , Numerische Mathematik, 130 (2015), pp. 467–496

work page 2015
[12]

Hu and X

X. Hu and X. Cheng , Acceleration of a two-grid method for eigenvalue problems , Mathematics of Computation, 80 (2011), pp. 1287–1301

work page 2011
[13]

Knyazev , Toward the optimal preconditioned eigensolver: locally op timal block precondi- tioned conjugate gradient method , SIAM J

A. Knyazev , Toward the optimal preconditioned eigensolver: locally op timal block precondi- tioned conjugate gradient method , SIAM J. Sci. Comput., 23 (2001), pp. 517–541

work page 2001
[14]

Knyazev and J

A. Knyazev and J. Osborn , New a priori FEM error estimates for eigenvalues , SIAM Journal on Numerical Analysis, 43 (2006), pp. 2647–2667

work page 2006
[15]

Li and L.-H

R.-C. Li and L.-H. Zhang , Convergence of the block Lanczos method for eigenvalue clus ters, Numerische Mathematik, 131 (2015), pp. 83–113

work page 2015
[16]

Liang, W

Q. Liang, W. W ang, and X. Xu , A two-level block preconditioned Jacobi–Davidson method for multiple and clustered eigenvalues of elliptic operato rs, SIAM Journal on Numerical Analysis, 62 (2024), pp. 998–1019

work page 2024
[17]

Lin and H

Q. Lin and H. Xie , A multi-level correction scheme for eigenvalue problems , Mathematics of Computation, 84 (2015), pp. 71–88

work page 2015
[18]

Liu and T

X. Liu and T. Vejchodsk ´y, Fully computable a posteriori error bounds for eigenfuncti ons, Numerische Mathematik, 152 (2022), pp. 183–221

work page 2022
[19]

Ovtchinnikov, Convergence estimates for preconditioned gradient subspa ce iteration eigen- solvers, Technical Report 1244, Preprints, Universitat Utrecht, ( 2002)

E. Ovtchinnikov, Convergence estimates for preconditioned gradient subspa ce iteration eigen- solvers, Technical Report 1244, Preprints, Universitat Utrecht, ( 2002)

work page 2002
[20]

Ovtchinnikov, Cluster robust error estimates for the Rayleigh-Ritz appro ximation

E. Ovtchinnikov, Cluster robust error estimates for the Rayleigh-Ritz appro ximation. I. Esti- mates for invariant subspaces , Linear Algebra and its Applications, 415 (2006), pp. 167–1 87

work page 2006
[21]

Ovtchinnikov , Cluster robust error estimates for the Rayleigh-Ritz appro ximation

E. Ovtchinnikov , Cluster robust error estimates for the Rayleigh-Ritz appro ximation. II. Es- timates for eigenvalues , Linear Algebra and its Applications, 415 (2006), pp. 188–2 09

work page 2006
[22]

Ovtchinnikov , Cluster robustness of preconditioned gradient subspace it eration eigen- solvers, Linear Algebra and its Applications, 415 (2006), pp

E. Ovtchinnikov , Cluster robustness of preconditioned gradient subspace it eration eigen- solvers, Linear Algebra and its Applications, 415 (2006), pp. 140–1 66

work page 2006
[23]

R ¨ohrig-Z¨ollner, J

M. R ¨ohrig-Z¨ollner, J. Thies, M. Kreutzer, A. Alvermann, A. Pieper, A. Baserm ann, G. Hager, G. Wellein, and H. Fehske , Increasing the performance of the Jacobi- Davidson method by blocking , SIAM Journal on Scientiﬁc Computing, 37 (2015), pp. C697– C722

work page 2015
[24]

Saad , Numerical methods for large eigenvalue problems , vol

Y. Saad , Numerical methods for large eigenvalue problems , vol. 66 of Classics in Applied Mathematics, Society for Industrial and Applied Mathemati cs (SIAM), Philadelphia, PA, 2011

work page 2011
[25]

Schwetlick and U

H. Schwetlick and U. Kandler , Block Newton method and block Rayleigh quotient iteration for computing invariant subspaces of general complex matri ces, Linear Algebra and its Applications, 526 (2017), pp. 60–94

work page 2017
[26]

G. L. G. Sleijpen and H. A. V an der Vorst , A Jacobi-Davidson iteration method for linear eigenvalue problems , SIAM Review, 42 (2000), pp. 267–293

work page 2000
[27]

Toselli and O

A. Toselli and O. Widlund , Domain Decomposition Methods-Algorithms and Theory , vol. 34 of Springer Series in Computational Mathematics, Springer -Verlag, Berlin, 2005

work page 2005
[28]

W ang and X

W. W ang and X. Xu , A two-level overlapping hybrid domain decomposition metho d for eigen- value problems , SIAM Journal on Numerical Analysis, 56 (2018), pp. 344–368

work page 2018
[29]

W ang and X

W. W ang and X. Xu , On the convergence of a two-level preconditioned Jacobi-Da vidson method for eigenvalue problems , Mathematics of Computation, 88 (2019), pp. 2295–2324

work page 2019
[30]

Xu and A

J. Xu and A. Zhou , A two-grid discretization scheme for eigenvalue problems , Mathematics of Computation, 70 (2001), pp. 17–25

work page 2001
[31]

Xu and A

J. Xu and A. Zhou , Local and parallel ﬁnite element algorithms for eigenvalue problems, Acta Mathematicae Applicatae Sinica. English Series, 18 (2002) , pp. 185–200

work page 2002
[32]

Yang and H

Y. Yang and H. Bi , Two-grid ﬁnite element discretization schemes based on shi fted-inverse power method for elliptic eigenvalue problems , SIAM Journal on Numerical Analysis, 49 (2011), pp. 1602–1624

work page 2011
[33]

Zhou and K

M. Zhou and K. Neymeyr , Cluster robust estimates for block gradient-type eigensol vers, Math- ematics of Computation, 88 (2019), pp. 2737–2765. This manuscript is for review purposes only

work page 2019

[1] [1]

I. M. Babu ˇska and J. Osborn , Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems , Mathematics of Computation, 52 (1989), pp. 275–297

work page 1989

[2] [2]

Boffi , Finite element approximation of eigenvalue problems , Acta Numerica, 19 (2010), pp

D. Boffi , Finite element approximation of eigenvalue problems , Acta Numerica, 19 (2010), pp. 1–120

work page 2010

[3] [3]

Boffi, D

D. Boffi, D. Gallistl, F. Gardini, and L. Gastaldi , Optimal convergence of adaptive FEM for eigenvalue clusters in mixed form , Mathematics of Computation, 86 (2017), pp. 2213– 2237

work page 2017

[4] [4]

Bonito and A

A. Bonito and A. Demlow , Convergence and optimality of higher-order adaptive ﬁnite el- ement methods for eigenvalue clusters , SIAM Journal on Numerical Analysis, 54 (2016), pp. 2379–2388

work page 2016

[5] [5]

Canc `es, G

E. Canc `es, G. Dusson, Y. Maday, B. Stamm, and M. Vohral ´ık, Guaranteed a posteri- ori bounds for eigenvalues and eigenvectors: multipliciti es and clusters , Mathematics of Computation, 89 (2020), pp. 2563–2611

work page 2020

[6] [6]

W. Chen, N. Shao, and X. Xu , A locally optimal preconditioned Newton-Schur method for symmetric elliptic eigenvalue problems , Mathematics of Computation, 92 (2023), pp. 2655– 2684. This manuscript is for review purposes only. 20 QIGANG LIANG AND XUEJUN XU

work page 2023

[7] [7]

X. Dai, L. He, and A. Zhou , Convergence and quasi-optimal complexity of adaptive ﬁnit e element computations for multiple eigenvalues , IMA Journal of Numerical Analysis, 35 (2015), pp. 1934–1977

work page 2015

[8] [8]

X. Dai, Y. Li, B. Yang, and A. Zhou , Numerical analysis of the parallel orbital-updating ap- proach for eigenvalue problems , SIAM Journal on Numerical Analysis, 63 (2025), pp. 1886– 1908

work page 2025

[9] [9]

X. Dai, J. Xu, and A. Zhou , Convergence and optimal complexity of adaptive ﬁnite eleme nt eigenvalue computations , Numerische Mathematik, 110 (2008), pp. 313–355

work page 2008

[10] [10]

Gallistl , Adaptive nonconforming ﬁnite element approximation of eig envalue clusters , Computational Methods in Applied Mathematics, 14 (2014), p p

D. Gallistl , Adaptive nonconforming ﬁnite element approximation of eig envalue clusters , Computational Methods in Applied Mathematics, 14 (2014), p p. 509–535

work page 2014

[11] [11]

Gallistl , An optimal adaptive FEM for eigenvalue clusters , Numerische Mathematik, 130 (2015), pp

D. Gallistl , An optimal adaptive FEM for eigenvalue clusters , Numerische Mathematik, 130 (2015), pp. 467–496

work page 2015

[12] [12]

Hu and X

X. Hu and X. Cheng , Acceleration of a two-grid method for eigenvalue problems , Mathematics of Computation, 80 (2011), pp. 1287–1301

work page 2011

[13] [13]

Knyazev , Toward the optimal preconditioned eigensolver: locally op timal block precondi- tioned conjugate gradient method , SIAM J

A. Knyazev , Toward the optimal preconditioned eigensolver: locally op timal block precondi- tioned conjugate gradient method , SIAM J. Sci. Comput., 23 (2001), pp. 517–541

work page 2001

[14] [14]

Knyazev and J

A. Knyazev and J. Osborn , New a priori FEM error estimates for eigenvalues , SIAM Journal on Numerical Analysis, 43 (2006), pp. 2647–2667

work page 2006

[15] [15]

Li and L.-H

R.-C. Li and L.-H. Zhang , Convergence of the block Lanczos method for eigenvalue clus ters, Numerische Mathematik, 131 (2015), pp. 83–113

work page 2015

[16] [16]

Liang, W

Q. Liang, W. W ang, and X. Xu , A two-level block preconditioned Jacobi–Davidson method for multiple and clustered eigenvalues of elliptic operato rs, SIAM Journal on Numerical Analysis, 62 (2024), pp. 998–1019

work page 2024

[17] [17]

Lin and H

Q. Lin and H. Xie , A multi-level correction scheme for eigenvalue problems , Mathematics of Computation, 84 (2015), pp. 71–88

work page 2015

[18] [18]

Liu and T

X. Liu and T. Vejchodsk ´y, Fully computable a posteriori error bounds for eigenfuncti ons, Numerische Mathematik, 152 (2022), pp. 183–221

work page 2022

[19] [19]

Ovtchinnikov, Convergence estimates for preconditioned gradient subspa ce iteration eigen- solvers, Technical Report 1244, Preprints, Universitat Utrecht, ( 2002)

E. Ovtchinnikov, Convergence estimates for preconditioned gradient subspa ce iteration eigen- solvers, Technical Report 1244, Preprints, Universitat Utrecht, ( 2002)

work page 2002

[20] [20]

Ovtchinnikov, Cluster robust error estimates for the Rayleigh-Ritz appro ximation

E. Ovtchinnikov, Cluster robust error estimates for the Rayleigh-Ritz appro ximation. I. Esti- mates for invariant subspaces , Linear Algebra and its Applications, 415 (2006), pp. 167–1 87

work page 2006

[21] [21]

Ovtchinnikov , Cluster robust error estimates for the Rayleigh-Ritz appro ximation

E. Ovtchinnikov , Cluster robust error estimates for the Rayleigh-Ritz appro ximation. II. Es- timates for eigenvalues , Linear Algebra and its Applications, 415 (2006), pp. 188–2 09

work page 2006

[22] [22]

Ovtchinnikov , Cluster robustness of preconditioned gradient subspace it eration eigen- solvers, Linear Algebra and its Applications, 415 (2006), pp

E. Ovtchinnikov , Cluster robustness of preconditioned gradient subspace it eration eigen- solvers, Linear Algebra and its Applications, 415 (2006), pp. 140–1 66

work page 2006

[23] [23]

R ¨ohrig-Z¨ollner, J

M. R ¨ohrig-Z¨ollner, J. Thies, M. Kreutzer, A. Alvermann, A. Pieper, A. Baserm ann, G. Hager, G. Wellein, and H. Fehske , Increasing the performance of the Jacobi- Davidson method by blocking , SIAM Journal on Scientiﬁc Computing, 37 (2015), pp. C697– C722

work page 2015

[24] [24]

Saad , Numerical methods for large eigenvalue problems , vol

Y. Saad , Numerical methods for large eigenvalue problems , vol. 66 of Classics in Applied Mathematics, Society for Industrial and Applied Mathemati cs (SIAM), Philadelphia, PA, 2011

work page 2011

[25] [25]

Schwetlick and U

H. Schwetlick and U. Kandler , Block Newton method and block Rayleigh quotient iteration for computing invariant subspaces of general complex matri ces, Linear Algebra and its Applications, 526 (2017), pp. 60–94

work page 2017

[26] [26]

G. L. G. Sleijpen and H. A. V an der Vorst , A Jacobi-Davidson iteration method for linear eigenvalue problems , SIAM Review, 42 (2000), pp. 267–293

work page 2000

[27] [27]

Toselli and O

A. Toselli and O. Widlund , Domain Decomposition Methods-Algorithms and Theory , vol. 34 of Springer Series in Computational Mathematics, Springer -Verlag, Berlin, 2005

work page 2005

[28] [28]

W ang and X

W. W ang and X. Xu , A two-level overlapping hybrid domain decomposition metho d for eigen- value problems , SIAM Journal on Numerical Analysis, 56 (2018), pp. 344–368

work page 2018

[29] [29]

W ang and X

W. W ang and X. Xu , On the convergence of a two-level preconditioned Jacobi-Da vidson method for eigenvalue problems , Mathematics of Computation, 88 (2019), pp. 2295–2324

work page 2019

[30] [30]

Xu and A

J. Xu and A. Zhou , A two-grid discretization scheme for eigenvalue problems , Mathematics of Computation, 70 (2001), pp. 17–25

work page 2001

[31] [31]

Xu and A

J. Xu and A. Zhou , Local and parallel ﬁnite element algorithms for eigenvalue problems, Acta Mathematicae Applicatae Sinica. English Series, 18 (2002) , pp. 185–200

work page 2002

[32] [32]

Yang and H

Y. Yang and H. Bi , Two-grid ﬁnite element discretization schemes based on shi fted-inverse power method for elliptic eigenvalue problems , SIAM Journal on Numerical Analysis, 49 (2011), pp. 1602–1624

work page 2011

[33] [33]

Zhou and K

M. Zhou and K. Neymeyr , Cluster robust estimates for block gradient-type eigensol vers, Math- ematics of Computation, 88 (2019), pp. 2737–2765. This manuscript is for review purposes only

work page 2019