Pith. sign in

REVIEW 2 major objections 4 minor 24 references

Galerkin approximation of holomorphic eigenvalue problems: weak T-coercivity and T-compatibility

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Weak T-coercivity plus T-compatibility yields convergence for Galerkin eigenvalue approximations.

desk verdict A genuinely useful sufficient condition for regularity of Galerkin eigenvalue approximations; the main theorem is sound, but the bridge to Karma's convergence and error estimates has a proof gap that should be fixed before publication. read the letter →

arxiv 1908.05029 v1 pith:2J6WCI2U submitted 2019-08-14 math.NA cs.NAmath.SP

Martin Halla This is my paper · ORCID
classification math.NAcs.NAmath.SP MSC 47J1065H1765N25
keywords holomorphiceigenvalueproblemnonlinearGalerkinapproximationT-coercivityregularityofapproximationsspectralpollutionFredholmoperatorfunctionsdiscretescheme
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves a two-condition recipe for Galerkin approximations of eigenvalue problems in which the operator depends holomorphically on the spectral parameter and is Fredholm: a continuous condition and a discrete condition. On the continuous side, the operator family must be weakly T-coercive, meaning a fixed bijective $T$ makes $T^*A$ a compact perturbation of a coercive operator; this secures Fredholmness with index zero. On the discrete side, the Galerkin approximations must be $T$-compatible, meaning each discrete operator is Fredholm with index zero and there are discrete operators $T_n$ converging to $T$ in the discrete norm. Together these conditions imply that the approximations are regular, and the standard theory of regular approximations then yields convergence of eigenvalues, no spectral pollution, and eigenvalue error bounds of order $(\delta_n \delta_n^*)^{1/\kappa}$. The result matters because it covers problems whose operator values are not 'coercive plus compact', such as sign-changing coefficient models and boundary integral eigenvalue formulations that are nonlinear in the eigenvalue parameter.

What carries the argument

The machinery has three pieces. Weak $T$-coercivity: $A$ is weakly $T$-coercive if there is a bijective $T \in L(X)$ and a compact $K$ such that $T^*A + K$ is coercive; this is the continuous condition that guarantees $A$ is Fredholm with index zero. $T$-compatibility: the discrete operators $A_n$ are Fredholm with index zero, and there exist index-zero Fredholm operators $T_n$ on $X_n$ such that $\|T - T_n\|_n \to 0$, where the discrete norm $\|\cdot\|_n$ is the operator norm on the Galerkin subspace $X_n$. The proof's engine is Lemma 1.7, which uses these two conditions to show that $A_n + P_n T^{-*}K|_{X_n}$ is invertible with a uniform bound on the inverse; Theorem 1.8 then converts compactness of $(A_n u_n)$ into compactness of $(u_n)$. The operator-function version carries the same argument pointwise in $\lambda$, and Lemma 2.6 extends an eigenspace error estimate to the general holomorphic Fredholm setting.

What would settle it

Because Theorem 1.8 is a universal statement, the direct falsifier is a counterexample: an operator $A$ and a Galerkin sequence satisfying weak $T$-coercivity and $T$-compatibility, together with a bounded sequence $u_n \in X_n$ such that $A_n u_n$ is compact but $u_n$ has no convergent subsequence. Short of a counterexample, one can test the necessity of the discrete Fredholm condition by comparing $T$-compatible and non-$T$-compatible discretizations of the same problem and checking whether spurious eigenvalues appear.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.8: if $A$ is weakly $T$-coercive and the Galerkin approximations $A_n := P_n A|_{X_n}$ are $T$-compatible, then $(A_n)$ is regular. Regularity is a compactness condition: any bounded sequence $(u_n)$ of discrete vectors for which $(A_n u_n)$ is compact must itself have a convergent subsequence. The paper shows that $T$-compatibility makes $A_n + P_n T^{-*}K|_{X_n}$ uniformly invertible for large $n$, and from that, compactness of $(A_n u_n)$ forces convergence of $(u_n)$. With regularity in hand, the established theory of regular approximations of holomorphic Fredholm operator functions yields the full convergence package: every eigenvalue is attained by discrete eigenvalues, there is no spectral pollution, eigenpairs cluster at genuine eigenpairs, the discrete family is stable away from the spectrum, the dimensions of generalized eigenspaces are preserved inside contours, and eigenvalue errors obey $|\lambda_0 - \lambda_n| \leq c(\delta_n \delta_n^*)^{1/\kappa}$, where $\kappa$ is the maximal Jordan chain length and $\delta_n, \delta_n^*$ measure how well the discrete spaces approximate the eigenspace and its adjoint. The paper also generalizes an existing eigenspace error estimate to arbitrary holomorphic Fredholm operator functions.

Load-bearing premise

The load-bearing premise is that each discrete operator $A_n(\lambda)$ is Fredholm with index zero and that index-zero Fredholm operators $T_n$ exist on the discrete spaces and converge to $T$ in the discrete norm; this must be checked for each discretization and does not follow automatically from Galerkin projection.

Editorial extensions

If this is right

  • For any weakly $T$-coercive holomorphic Fredholm eigenvalue problem, a Galerkin scheme that is $T$-compatible automatically produces a regular approximation, so no spurious eigenvalues can appear and every true eigenvalue is approximated by discrete eigenvalues.
  • The eigenvalue error is bounded by $c(\delta_n \delta_n^*)^{1/\kappa}$, where $\delta_n$ and $\delta_n^*$ are best-approximation errors of the eigenspace and its adjoint and $\kappa$ is the maximal Jordan chain length; improving either discrete space improves the eigenvalue rate.
  • The theory covers operator families that are not 'coercive plus compact', including sign-changing coefficient models and boundary integral formulations whose dependence on the spectral parameter is nonlinear.
  • Normalized discrete eigenvectors satisfy an explicit bound combining the eigenvalue error and the best-approximation error of the true eigenspace, which is new for general holomorphic Fredholm operator functions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The hypothesis that is hardest to verify in practice is not weak $T$-coercivity but the discrete part of $T$-compatibility: each $A_n(\lambda)$ must be Fredholm of index zero, and natural mixed or boundary element discretizations can fail this even when the continuous operator satisfies the framework.
  • Because the definition of $T$-compatibility only asks for discrete-norm convergence of $T_n$ to $T$, it gives a practical recipe: build an explicit stabilizer $T$, discretize it, and check the discrete norm; $T$-invariant subspaces are not required.
  • A natural numerical test is to compute eigenvalues for a sign-changing coefficient problem with and without a $T$-compatible discretization; the framework predicts that only the $T$-compatible scheme avoids pollution and attains the $(\delta_n \delta_n^*)^{1/\kappa}$ rate.
  • The Hilbert-space formulation suggests a Banach-space or non-orthogonal projection extension, since the discrete norm and regularity arguments do not fundamentally rely on orthogonality.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies Galerkin approximations of holomorphic Fredholm operator eigenvalue problems without the usual 'coercive+compact' structure. It introduces weak T-coercivity and T-compatibility: a bounded operator A is weakly T-coercive if T^*A is a compact perturbation of a coercive operator, and a Galerkin approximation is T-compatible if there exist index-zero Fredholm operators T_n on X_n converging to T in the discrete norm. Theorem 1.8 proves that T-compatible Galerkin approximations of weakly T-coercive operators are regular in the sense that compactness of (A_n u_n) forces compactness of (u_n). Section 2 extends this to operator functions and combines it with Karma's spectral approximation theory: Proposition 2.7 lists no spectral pollution, eigenvalue convergence, stability on compact resolvent sets, eigenvalue/eigenspace error bounds, and Lemma 2.6 states a new eigenspace error estimate. Corollary 2.8 applies the framework to weakly T(·)-coercive operator functions. The abstract advertises applications to Hohage-Nannen infinite element methods and Bonnet-Ben Dhia-Carvalho-Ciarlet finite element methods.

Significance. The central idea is useful and the main theorem is cleanly proved. The proof of Theorem 1.8 is based on a uniform inf-sup estimate for the perturbed Galerkin operators A_n + P_n T^{-*}K (Lemma 1.7), and the argument is sound. If the application to Karma's theory is made rigorous, the paper would provide a practical criterion for spectral exactness and error estimates in non-coercive settings, improving results of Hohage-Nannen and Unger. The paper is honest about the main difficulty: verifying that the discrete operators are Fredholm index zero and that the operators T_n exist is often the hard part in practice. However, the manuscript is not yet a complete proof of the advertised convergence claims because of a gap in the proof of Proposition 2.7.

major comments (2)
  1. [§2, Proposition 2.7, proof] The proof of Proposition 2.7 states: 'Assumption b2 follows from Lemma 1.7 (at least for sufficiently large n).' But Lemma 1.7 is proved under the hypotheses that A is weakly T-coercive and that (A_n) is T-compatible; Proposition 2.7 assumes neither. The assumptions of Proposition 2.7 are only holomorphy, Fredholm index zero of A(λ) and A_n(λ), non-empty resolvent, pointwise convergence of the projections, and regularity of (A_n(·)) in the sense of Definition 2.3. Consequently, the verification of Karma's assumption b2 is invalid as written, and the claims i)–vii) are not established by the argument presented. The authors should either add the missing weak T-coercivity/T-compatibility hypotheses to Proposition 2.7 (at the cost of making Corollary 2.8 a tautology) or replace this step by a direct argument that the stated regularity hypothesis implies the relevant assumption of Karma's theorem for Galerkin schemes.
  2. [§2, Lemma 2.6, proof] The proof of Lemma 2.6 is a single sentence: it asserts that the needed result 'already follows from [18, Theorem 2 ii)]'. Since Lemma 2.6 is new (an improvement of Unger's Theorem 4.3.7) and is the basis for the eigenspace error estimate (vii), the proof should spell out why the hypotheses of [18, Theorem 2 ii)] are satisfied and how that theorem yields the bound (11). As written, this step is a citation rather than a proof, and the eigenspace error claim is therefore not fully supported.
minor comments (4)
  1. [§2, Theorem 2.4, proof] The proof of Theorem 2.4 says 'Follows from Theorem 2.4', which is circular; it should refer to Theorem 1.8.
  2. [§1, Lemma 1.7, proof] The displayed inf-sup chain contains a misplaced parenthesis: '|⟨((A + T −∗K)un, T v n⟩X |' should read '|⟨(A+T^{-*}K)u_n, Tv_n angle_X|'.
  3. [§1, Theorem 1.8, proof] In the first sentence of the proof, 'let (u_n ∈ L(X_n))' should be 'let (u_n ∈ X_n)'.
  4. [§2, Definition 2.2] The phrase 'T(λ) compatible' is missing a hyphen; it should be 'T(λ)-compatible'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central theorem is a new conditional result proved from definitions, and the cited Karma results are external, not self-citations.

full rationale

The paper's main claim is Theorem 1.8: weak T-coercivity plus T-compatibility of a Galerkin approximation implies regularity. This is not circular. T-compatibility (Definition 1.3) requires that the discrete operators An are Fredholm of index zero and that there are index-zero Fredholm operators Tn converging to T in the discrete norm. Regularity (Definition 1.5) is a different property: bounded discrete sequences with compact images must themselves be compact. The proof of Theorem 1.8 uses Lemma 1.7 to get uniform invertibility of An + Pn T^{-*} K | Xn and then a standard compact-perturbation argument with pointwise convergence of projections. There are no fitted parameters, and the conclusion is not contained in the premise by definition. The generalization to operator functions (Theorem 2.4) is a direct pointwise application of Theorem 1.8, and the typo in its proof ('follows from Theorem 2.4') does not make the argument circular because Theorem 1.8 is independently proved. Proposition 2.7's proof has a gap: it appeals to Lemma 1.7 to verify Karma's assumption b2 even though Proposition 2.7 does not assume weak T-coercivity or T-compatibility. This is a correctness or hypothesis issue, not circularity: Lemma 1.7 is not equivalent to the conclusion, and the relevant hypotheses are present when Corollary 2.8 applies Theorem 2.4. There are no load-bearing self-citations by the author; the cited external results by Karma, Hohage-Nannen, Bonnet-Ben Dhia-Carvalho-Ciarlet, and Unger are independent prior work. The paper is self-contained relative to its stated assumptions, so the appropriate finding is no significant circularity with score 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper's results are conditional theorems; the ledger records background results and structural assumptions. No fitted parameters are present, and the new concepts (weak T-coercivity, T-compatibility) are definitions rather than empirical entities.

assumptions (5)
  • standard math Karma's theorems on approximation of holomorphic Fredholm operator functions ([18], [19]) are valid and applicable.
    Proposition 2.7 and Corollary 2.8 derive convergence and error bounds directly from Karma 1996 I/II; the paper does not reprove them.
  • domain assumption The orthogonal projections Pn onto Xn converge pointwise to the identity on X.
    Assumed at the start of Section 1 and used to verify the discrete approximation scheme axioms (Lemma 2.5).
  • domain assumption The operator function A(.) : Lambda -> L(X) is holomorphic Fredholm with non-empty resolvent set.
    Standard setting for eigenvalue approximation; stated as assumption in Proposition 2.7 and Corollary 2.8.
  • domain assumption For a weakly T-coercive A, there exists a compact operator K such that T* A + K is coercive.
    This is the defining property of weak T-coercivity, used in Lemma 1.7 and Theorem 1.8.
  • standard math Lax-Milgram lemma and standard Fredholm theory (bounded compact perturbation of Fredholm operator is Fredholm of same index).
    Used implicitly in Lemma 1.7 to conclude invertibility and index properties.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Galerkin approximation of holomorphic eigenvalue problems: weak T-coercivity and T-compatibility." pith.science (2026). https://pith.science/paper/2J6WCI2U

@misc{pith2026190805029,
  author       = {Pith},
  title        = {Pith review of: Galerkin approximation of holomorphic eigenvalue problems: weak T-coercivity and T-compatibility},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2J6WCI2U}},
  note         = {Machine review of arXiv:1908.05029}
}
read the original abstract

We consider Galerkin approximations of holomorphic Fredholm operator eigenvalue problems for which the operator values don't have the structure "coercive+compact". In this case the regularity (in sense of [O. Karma, Numer. Funct. Anal. Optim. 17 (1996)]) of Galerkin approximations is not unconditionally satisfied and the question of convergence is delicate. We report a technique to prove regularity of approximations which is applicable to a wide range of eigenvalue problems. In particular, we introduce the concepts of weak T-coercivity and T-compatibility and prove that for weakly T-coercive operators, T-compatibility of Galerkin approximations implies their regularity. Our framework immediately improves the results of [T. Hohage, L. Nannen, BIT 55(1) (2015)], is immediately applicable to analyze approximations of eigenvalue problems related to [A.-S. Bonnet-Ben Dhia, C. Carvalho, P. Ciarlet, Num. Math. 138(4) (2018)] and is already applied in [G. Unger, preprint (2017)].

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

24 extracted references · 19 canonical work pages

  1. [1]

    P. M. Anselone and M. L. Treuden, Regular operator approximation theory , Pacific J. Math. 120 (1985), no. 2, 257–268. MR 810769 (87a:47017)

    work page 1985

  2. [2]

    Arnold, Richard S

    Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus: from Hodge theory to numerical stability , Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 2, 281–354. MR 2594630 (2011f:58005)

    2010

  3. [3]

    Daniele Boffi, Franco Brezzi, and Lucia Gastaldi, On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form , Math. Comp. 69 (2000), no. 229, 121–140. MR 1642801 (2000i:65175)

    work page 2000

  4. [4]

    4, 801–838

    Anne-Sophie Bonnet-BenDhia, Camille Carvalho, and Patr ick Ciarlet, Mesh requirements for the finite element approximation of problems with sign-c hanging coefficients , Numerische Mathematik 138 (2018), no. 4, 801–838

    work page 2018

  5. [5]

    Anne-Sophie Bonnet-BenDhia, Patrick Ciarlet, and Carlo Maria Zw¨ olf,Time harmonic wave diffraction problems in materials with sign-shifting coeffic ients, Journal of Computational and Applied Mathematics 234(6) (2010), 1912–1919

    2010

  6. [6]

    Annalisa Buffa, Remarks on the discretization of some noncoercive operator with applications to heterogeneous maxwell equations , SIAM Journal on Numerical Analysis 43 (2005), no. 1, 1–18

    2005

  7. [7]

    thesis, 12 2015

    Camille Carvalho, Etude math´ ematique et num´ erique de structures plasmoniq ue avec des coins., Ph.D. thesis, 12 2015

    work page 2015

  8. [8]

    Lucas Chesnel, Interior transmission eigenvalue problem for maxwell’s eq uations: the T- coercivity as an alternative approach , Inverse problems 28 (2012), 065005. 9. , ´Etude de quelques probl` emes de transmission avec changeme nt de signe. application aux m´ etamat´ eriaux., Ph.D. thesis, 10 2012

    work page 2012

Show all 24 references
  1. [10]

    Jean Descloux, Nabil Nassif, and Jacques Rappaz, On spectral approximation. part 1. the prob- lem of convergence , ESAIM: Mathematical Modelling and Numerical Analysis - Mo d´ elisation Math´ ematique et Analyse Num´ erique12 (1978), no. 2, 97–112 (eng)

    1978

  2. [11]

    , On spectral approximation. part 2. error estimates for the g alerkin method, ESAIM: Mathematical Modelling and Numerical Analysis - Mod´ elisa tion Math´ ematique et Analyse Num´ erique12 (1978), no. 2, 113–119 (eng)

    1978

  3. [12]

    Kaasho ek, Classes of linear operators

    Israel Gohberg, Seymour Goldberg, and Marinus A. Kaasho ek, Classes of linear operators. Vol. I , Operator Theory: Advances and Applications, vol. 49, Birk h¨ auser Verlag, Basel, 1990. MR 1130394 (93d:47002)

    1990

  4. [13]

    192, Bir kh¨ auser Verlag, Basel, 2009, Methods from complex analysis in several variables

    Israel Gohberg and J¨ urgen Leiterer, Holomorphic operator functions of one variable and applications, Operator Theory: Advances and Applications, vol. 192, Bir kh¨ auser Verlag, Basel, 2009, Methods from complex analysis in several variables. M R 2527384 (2010i:47035)

    2009

  5. [14]

    I, II , Math

    Rolf Dieter Grigorieff, Zur Theorie linearer approximationsregul¨ arer Operatoren. I, II , Math. Nachr. 55 (1973), 233–249; ibid. 55 (1973), 251–263. MR 0348533 (50 #1 031)

    1973

  6. [15]

    3, 245–271 (German)

    Rolf Dieter Grigorieff and Hansgeorg Jeggle, Approximation von Eigenwertproblemen bei nichtlinearer Parameterabh¨ angigkeit, manuscripta mathematica 10 (1973), no. 3, 245–271 (German)

    1973

  7. [16]

    1, 215–254 (English)

    Thorsten Hohage and Lothar Nannen, Convergence of infinite element methods for scalar waveguide problems, BIT Numerical Mathematics 55 (2015), no. 1, 215–254 (English)

    2015

  8. [17]

    Hansgeorg Jeggle and W olfgang W endland, On the discrete approximation of eigenvalue problems with holomorphic parameter dependence , Proc. Roy. Soc. Edinburgh Sect. A 78 (1977/78), no. 1-2, 1–29. MR 0474794 (57 #14427)

    1977

  9. [18]

    I , Numer

    Otto Karma, Approximation in eigenvalue problems for holomorphic Fred holm operator func- tions. I , Numer. Funct. Anal. Optim. 17 (1996), no. 3-4, 365–387. MR 1393166 (98e:47030) WEAK T-COERCIVITY AND T-COMPATIBILITY 11

    1996

  10. [19]

    , Approximation in eigenvalue problems for holomorphic Fred holm operator functions. II. (Convergence rate), Numer. Funct. Anal. Optim. 17 (1996), no. 3-4, 389–408. MR 1393167 (98e:47031)

    1996

  11. [20]

    MR 1 729870

    Vladimir Kozlov and Vladimir Maz ′ya, Differential equations with operator coefficients with applications to boundary value problems for partial differe ntial equations , Springer Mono- graphs in Mathematics, Springer-Verlag, Berlin, 1999. MR 1 729870

    1999

  12. [21]

    I , Math

    Friedrich Stummel, Diskrete Konvergenz linearer Operatoren. I , Math. Ann. 190 (1970/71), 45–92. MR 0291870 (45 #959)

    1970

  13. [22]

    thesis, TU Graz, Graz, Austria, 2009

    Gerhard Unger, Analysis of boundary element methods for laplacian eigenva lue problems , Ph.D. thesis, TU Graz, Graz, Austria, 2009

    2009

  14. [23]

    Report 2017/2, Institute of Computational Mathemat ics, Graz University of Technology, 2017

    , Convergence analysis of a galerkin boundary element method for electromagnetic eigenvalue problems , Tech. Report 2017/2, Institute of Computational Mathemat ics, Graz University of Technology, 2017

    2017

  15. [24]

    G. M. Va ˘ ınikko and O. O. Karma, The rate of convergence of approximation methods for an eigenvalue problem in which the parameter occurs nonlinear ly, ˇZ. Vyˇ cisl. Mat. i Mat. Fiz. 14 (1974), 1393–1408, 1628. MR 0395217 (52 #16015)

    1974

  16. [25]

    Genadi Vainikko, Funktionalanalysis der Diskretisierungsmethoden , B. G. Teubner Verlag, Leipzig, 1976, Mit Englischen und Russischen Zusammenfass ungen, Teubner-Texte zur Math- ematik. MR 0468159 (57 #7997) E-mail address : martin.halla@protonmail.com

    1976

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.