Modal analysis of a domain decomposition method for Maxwell's equations in a waveguide

Antoine Tonnoir; Pierre-Henri Tournier; Victorita Dolean

arxiv: 2502.15548 · v3 · submitted 2025-02-21 · 🧮 math.NA · cs.NA· physics.comp-ph

Modal analysis of a domain decomposition method for Maxwell's equations in a waveguide

Victorita Dolean , Antoine Tonnoir , Pierre-Henri Tournier This is my paper

Pith reviewed 2026-05-23 02:33 UTC · model grok-4.3

classification 🧮 math.NA cs.NAphysics.comp-ph

keywords domain decompositionSchwarz methodMaxwell equationswaveguidesToeplitz matricesmodal analysisweak scalabilityPML

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The pith

Limiting spectra of Toeplitz matrices govern convergence of one-level Schwarz methods for Maxwell equations in general waveguides

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

This paper develops a theoretical framework to analyze the weak scalability of one-level Schwarz domain decomposition methods for time-harmonic Maxwell's equations in waveguides. It combines limiting spectrum analysis of Toeplitz matrices from strip-wise decompositions with modal decomposition of the solutions. The framework handles arbitrary cross-sections and both impedance and perfectly matched layer transmission conditions. Numerical experiments confirm that the limiting spectrum predicts practical convergence rates even with modest numbers of subdomains. The analysis also identifies domain decomposition parameters that make the iteration robust to the wave number.

Core claim

The convergence of the one-level Schwarz iteration for Maxwell's equations in strip-wise decomposed waveguides is governed by the limiting spectrum of associated Toeplitz matrices combined with the modal decomposition of the electromagnetic solutions, allowing extension to arbitrary cross-sections and impedance or PML transmission conditions.

What carries the argument

Limiting spectrum of Toeplitz matrices arising from strip-wise decomposition, analyzed together with modal decomposition of Maxwell solutions

Load-bearing premise

The limiting spectrum of the Toeplitz matrices from strip-wise decomposition together with modal decomposition continues to govern convergence of the one-level Schwarz iteration for arbitrary cross-sections and impedance or PML conditions.

What would settle it

A numerical computation of the actual iteration operator spectrum for a waveguide with non-rectangular cross-section that deviates substantially from the predicted limiting spectrum.

read the original abstract

Time-harmonic wave propagation problems, especially those governed by Maxwell's equations, pose significant computational challenges due to the non-self-adjoint nature of the operators and the large, non-Hermitian linear systems resulting from discretization. Domain decomposition methods, particularly one-level Schwarz methods, offer a promising framework to tackle these challenges, with recent advancements showing the potential for weak scalability under certain conditions. In this paper, we analyze the weak scalability of one-level Schwarz methods for Maxwell's equations in strip-wise domain decompositions, focusing on waveguides with general cross sections and different types of transmission conditions such as impedance or perfectly matched layers (PMLs). By combining techniques from the limiting spectrum analysis of Toeplitz matrices and the modal decomposition of Maxwell's solutions, we provide a novel theoretical framework that extends previous work to more complex geometries and transmission conditions. Numerical experiments confirm that the limiting spectrum effectively predicts practical behavior even with a modest number of subdomains. Furthermore, we demonstrate that the one-level Schwarz method can achieve robustness with respect to the wave number under specific domain decomposition parameters, offering new insights into its applicability for large-scale electromagnetic wave problems.

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

Extends Toeplitz-modal analysis to general cross-sections and PML but the structural preservation for arbitrary sections is the part that needs checking.

read the letter

The paper combines limiting spectra of Toeplitz matrices from strip-wise decompositions with modal analysis of Maxwell solutions to study convergence of one-level Schwarz methods in waveguides. The new piece is the claimed extension to arbitrary cross-sections and to both impedance and PML transmission conditions, plus the observation that the method can be made robust to wave number with the right parameters. Numerical tests are said to match the predictions even when the number of subdomains is small. That is the concrete advance over earlier rectangular-section work. The numerics are presented as confirmation rather than the main result, which is fair for this kind of analysis paper. The soft spot is exactly the one the stress-test note flags: once the cross-section is arbitrary the transverse eigenmodes no longer separate in a way that automatically produces a clean block-Toeplitz structure along the propagation direction, and PML stretching perturbs the interface symbols further. The abstract does not show the extra steps that would let the old limiting-spectrum formulas carry over unchanged, so the central claim rests on whether those formulas survive without additional assumptions on the transverse operator. If the full paper supplies a derivation or a clear reduction that handles the general case, the gap closes; otherwise the theoretical handle is weaker than advertised. This is aimed at people who already work on domain decomposition for time-harmonic Maxwell problems and want an analytical handle on scalability. A reader in that niche can extract the parameter choices that give wavenumber robustness and the practical confirmation that the spectrum still predicts behavior with few subdomains. It is incremental rather than foundational, but the topic is practically relevant and the approach is a direct extension of existing techniques, so it deserves a serious referee who can check the derivations and the precise conditions under which the Toeplitz structure is preserved.

Referee Report

2 major / 1 minor

Summary. The paper claims to develop a novel theoretical framework for the weak scalability of one-level Schwarz domain decomposition methods applied to time-harmonic Maxwell's equations in waveguides. The framework combines limiting-spectrum analysis of Toeplitz matrices with modal decomposition of the solutions, extending prior results to waveguides with arbitrary cross-sections and to impedance or PML transmission conditions. Numerical experiments are said to confirm that the limiting spectrum predicts practical convergence behavior even for modest numbers of subdomains, and that robustness with respect to the wave number is achievable under suitable domain-decomposition parameters.

Significance. If the structural preservation of the limiting spectrum under arbitrary cross-sections and PML stretching can be rigorously established, the work would supply a useful analytic tool for predicting convergence of non-overlapping Schwarz methods in electromagnetic waveguide problems. The explicit combination of Toeplitz limiting spectra with modal analysis, together with the reported numerical confirmation for small subdomain counts, would constitute a concrete advance over purely numerical studies of the same methods.

major comments (2)

[Abstract / theoretical framework] Abstract (paragraph on the novel theoretical framework): the central claim requires that the limiting spectrum of the Toeplitz matrices arising from strip-wise decomposition, together with modal decomposition, continues to control one-level Schwarz convergence when the waveguide cross-section is arbitrary. For non-rectangular cross-sections the transverse eigenmodes generally lack the separability that produces a clean block-Toeplitz structure along the guide axis; the manuscript must show explicitly (e.g., in the derivation of the interface symbol) that the limiting-spectrum formulas survive this loss of separability without additional assumptions on the transverse operator.
[Abstract / theoretical framework] Abstract (paragraph on PML transmission conditions): PML stretching perturbs the interface symbols that enter the Toeplitz analysis. The manuscript must demonstrate that the limiting-spectrum formulas remain valid under this perturbation; otherwise the extension to PML conditions rests on an unverified structural assumption that is load-bearing for the claimed generality.

minor comments (1)

[Numerical experiments] The abstract states that numerical experiments confirm the predictions but supplies no information on mesh sizes, number of subdomains tested, or quantitative error measures; these details should be added to the numerical section for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major comments point by point below. In both cases we agree that additional explicit derivations are needed to fully substantiate the claims and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract / theoretical framework] Abstract (paragraph on the novel theoretical framework): the central claim requires that the limiting spectrum of the Toeplitz matrices arising from strip-wise decomposition, together with modal decomposition, continues to control one-level Schwarz convergence when the waveguide cross-section is arbitrary. For non-rectangular cross-sections the transverse eigenmodes generally lack the separability that produces a clean block-Toeplitz structure along the guide axis; the manuscript must show explicitly (e.g., in the derivation of the interface symbol) that the limiting-spectrum formulas survive this loss of separability without additional assumptions on the transverse operator.

Authors: The modal decomposition is performed with respect to the eigenfunctions of the transverse Maxwell operator defined on the (possibly non-rectangular) cross-section. These eigenfunctions form a complete basis that diagonalizes the transverse operator, thereby reducing the original problem to a countable set of independent one-dimensional modal problems along the waveguide axis. For each fixed mode the strip-wise decomposition produces a block-Toeplitz operator whose symbol depends only on the modal wave number and the chosen transmission condition; the transverse geometry enters solely through the modal eigenvalue and does not affect the longitudinal Toeplitz structure. Consequently the limiting-spectrum formulas derived for the one-dimensional modal problems apply directly. We will add an explicit derivation of the interface symbol in modal coordinates to the revised manuscript, confirming that the argument requires only the standard spectral properties of the transverse operator and no further separability assumptions. revision: yes
Referee: [Abstract / theoretical framework] Abstract (paragraph on PML transmission conditions): PML stretching perturbs the interface symbols that enter the Toeplitz analysis. The manuscript must demonstrate that the limiting-spectrum formulas remain valid under this perturbation; otherwise the extension to PML conditions rests on an unverified structural assumption that is load-bearing for the claimed generality.

Authors: We agree that the perturbation induced by PML stretching must be treated explicitly. In the modal setting the PML appears as a complex stretching of the longitudinal coordinate near the artificial interfaces, which modifies the transmission coefficients but leaves the block-Toeplitz character of the discrete operator intact. The limiting spectrum is obtained from the adjusted symbol that incorporates the stretched transmission parameters. We will insert a dedicated subsection deriving this modified symbol and verifying that the spectral bounds (and the resulting wave-number robustness under suitable parameter choices) continue to hold. This derivation will be placed immediately after the impedance-case analysis so that the extension to PML is fully rigorous. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation extends external Toeplitz limiting-spectrum and modal-analysis techniques

full rationale

The provided abstract and claims describe a framework obtained by combining standard limiting-spectrum analysis of Toeplitz matrices with modal decomposition of Maxwell solutions, then extending the combination to arbitrary cross-sections and impedance/PML transmission conditions. No equation, definition, or claim is shown that reduces a derived quantity to a fitted parameter taken from the same data, nor does any load-bearing step rest on a self-citation whose content is itself unverified within the paper. Numerical experiments are presented only as confirmation of the already-derived limiting spectrum, not as the source of the spectrum formulas. The derivation is therefore self-contained against the cited external mathematical techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard mathematical background rather than new postulates; no free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math Limiting spectrum analysis applies to the Toeplitz matrices generated by the strip-wise discretization of the waveguide operator
Invoked as the core analytical tool for predicting iteration behavior.
domain assumption Modal decomposition of Maxwell solutions remains valid for general cross sections under the chosen transmission conditions
Required to reduce the three-dimensional problem to a sequence of one-dimensional modal problems.

pith-pipeline@v0.9.0 · 5741 in / 1521 out tokens · 68199 ms · 2026-05-23T02:33:57.240999+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

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J. Bramble and J. Pasciak , Analysis of a finite pml approximation for the three dimensional time-harmonic maxwell and acoustic scattering problems , Mathematics of Computation, 76 (2007), pp. 597--614

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Brunet, V

R. Brunet, V. Dolean, and M. J. Gander , Natural domain decomposition algorithms for the solution of time-harmonic elastic waves , SIAM J. Sci. Comput., 42 (2020), pp. A3313--A3339

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E. Cances, Y. Maday, and B. Stamm , Domain decomposition for implicit solvation models , J. Chem. Phys., 139(5) (2013), https://doi.org/10.1063/1.4816767

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G. Ciaramella, M. Hassan, and B. Stamm , On the scalability of the S chwarz method , SMAI J. Comput. Math., 6 (2020), pp. 33--68, https://doi.org/10.5802/smai-jcm.61, https://doi.org/10.5802/smai-jcm.61

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Dolean, M

V. Dolean, M. J. Gander, and A. Kyriakis , Closed form optimized transmission conditions for complex diffusion with many subdomains , SIAM Journal on Scientific Computing, 45 (2023), pp. A829--A848

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Dolean, M

V. Dolean, M. J. Gander, S. Lanteri, J.-F. Lee, and Z. Peng , Effective transmission conditions for domain decomposition methods applied to the time-harmonic curl--curl M axwell's equations , J. Comput. Phys., 280 (2015), pp. 232--247, https://doi.org/10.1016/j.jcp.2014.09.024

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Dolean, P

V. Dolean, P. Jolivet, and F. Nataf , An Introduction to Domain Decomposition Methods: Algorithms, Theory, and Parallel Implementation , SIAM, Philadelphia, 2015, https://doi.org/10.1137/1.9781611974065.ch1

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Dolean, P

V. Dolean, P. Jolivet, P.-H. Tournier, and S. Operto , Iterative frequency-domain seismic wave solvers based on multi-level domain-decomposition preconditioners , in 82 ^ th Annual EAGE Meeting (Amsterdam), 2020. arXiv:2004.06309

work page arXiv 2020
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El Bouajaji , V

M. El Bouajaji , V. Dolean, M. J. Gander, and S. Lanteri , Optimized S chwarz methods for the time-harmonic M axwell equations with damping , SIAM J. Sci. Comput., 34 (2012), pp. A2048--A2071

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M. J. Gander, L. Halpern, and F. Magoul \`e s , An optimized S chwarz method with two-sided R obin transmission conditions for the H elmholtz equation , Internat. J. Numer. Methods Fluids, 55 (2007), pp. 163--175

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M. J. Gander and A. Tonnoir , Analysis of schwarz methods for convected helmholtz-like equations , SIAM Journal on Scientific Computing, 46 (2024), pp. A1--A22

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I. G. Graham, E. A. Spence, and J. Zou , Domain decomposition with local impedance conditions for the H elmholtz equation , SIAM J. Numer. Anal., 58 (2020), pp. 2515--2543

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H. Haddar, P. Joly, and H.-M. Nguyen , Generalized impedance boundary conditions for scattering problems from strongly absorbing obstacles: The case of maxwell's equations , Mathematical Models and Methods in Applied Sciences, 18 (2008), pp. 1787--1827

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V. Mattesi, M. Darbas, and C. Geuzaine , A high-order absorbing boundary condition for 2D time-harmonic elastodynamic scattering problems , Comput. Math. Appl., 77 (2019), pp. 1703--1721

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Royer, C

A. Royer, C. Geuzaine, E. B \'e chet, and A. Modave , A non-overlapping domain decomposition method with perfectly matched layer transmission conditions for the helmholtz equation , Computer Methods in Applied Mechanics and Engineering, 395 (2022), p. 115006

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write newline

" write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION not #0 #1 if FUNCTION and 'skip pop #0 if FUNCTION or pop #1 'skip if FUNCTION new.block.checka empty 'skip 'new.block if FUNCTION field.or.null duplicate empty pop "" 'skip ...

work page

[1] [1]

I. M. Babu s ka and S. A. Sauter , Is the pollution effect of the FEM avoidable for the H elmholtz equation considering high wave numbers? , SIAM J. Numer. Anal., 34 (1997), pp. 2392--2423

work page 1997

[2] [2]

Bonazzoli, V

M. Bonazzoli, V. Dolean, I. G. Graham, E. A. Spence, and P.-H. Tournier , Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption , Math. Comp., 88 (2019), pp. 2559--2604

work page 2019

[3] [3]

Bootland, S

N. Bootland, S. Borzooei, V. Dolean, and P.-H. Tournier , Numerical assessment of pml transmission conditions in a domain decomposition method for the helmholtz equation , in International Conference on Domain Decomposition Methods, Springer, 2022, pp. 445--453

work page 2022

[4] [4]

Bootland, V

N. Bootland, V. Dolean, A. Kyriakis, and J. Pestana , Analysis of parallel schwarz algorithms for time-harmonic problems using block toeplitz matrices , Electronic Transactions on Numerical Analysis, 55 (2022), pp. 112--141, https://doi.org/10.1553/etna_vol55s112

work page doi:10.1553/etna_vol55s112 2022

[5] [5]

Bramble and J

J. Bramble and J. Pasciak , Analysis of a finite pml approximation for the three dimensional time-harmonic maxwell and acoustic scattering problems , Mathematics of Computation, 76 (2007), pp. 597--614

work page 2007

[6] [6]

Brunet, V

R. Brunet, V. Dolean, and M. J. Gander , Natural domain decomposition algorithms for the solution of time-harmonic elastic waves , SIAM J. Sci. Comput., 42 (2020), pp. A3313--A3339

work page 2020

[7] [7]

Cances, Y

E. Cances, Y. Maday, and B. Stamm , Domain decomposition for implicit solvation models , J. Chem. Phys., 139(5) (2013), https://doi.org/10.1063/1.4816767

work page doi:10.1063/1.4816767 2013

[8] [8]

Ciaramella and M

G. Ciaramella and M. J. Gander , Analysis of the parallel S chwarz method for growing chains of fixed-sized subdomains: P art I , SIAM J. Numer. Anal., 55 (2017), pp. 1330--1356, https://doi.org/10.1137/16M1065215, https://doi.org/10.1137/16M1065215

work page doi:10.1137/16m1065215 2017

[9] [9]

Ciaramella, M

G. Ciaramella, M. Hassan, and B. Stamm , On the scalability of the S chwarz method , SMAI J. Comput. Math., 6 (2020), pp. 33--68, https://doi.org/10.5802/smai-jcm.61, https://doi.org/10.5802/smai-jcm.61

work page doi:10.5802/smai-jcm.61 2020

[10] [10]

A.-S. B.-B. Dhia and \'E . Lun \'e ville , Propagation dans les guides d'ondes , (2021)

work page 2021

[11] [11]

Dolean, M

V. Dolean, M. J. Gander, and L. Gerardo-Giorda , Optimized S chwarz methods for M axwell's equations , SIAM J. Sci. Comput., 31 (2009), pp. 2193--2213

work page 2009

[12] [12]

Dolean, M

V. Dolean, M. J. Gander, and A. Kyriakis , Closed form optimized transmission conditions for complex diffusion with many subdomains , SIAM Journal on Scientific Computing, 45 (2023), pp. A829--A848

work page 2023

[13] [13]

Dolean, M

V. Dolean, M. J. Gander, S. Lanteri, J.-F. Lee, and Z. Peng , Effective transmission conditions for domain decomposition methods applied to the time-harmonic curl--curl M axwell's equations , J. Comput. Phys., 280 (2015), pp. 232--247, https://doi.org/10.1016/j.jcp.2014.09.024

work page doi:10.1016/j.jcp.2014.09.024 2015

[14] [14]

Dolean, P

V. Dolean, P. Jolivet, and F. Nataf , An Introduction to Domain Decomposition Methods: Algorithms, Theory, and Parallel Implementation , SIAM, Philadelphia, 2015, https://doi.org/10.1137/1.9781611974065.ch1

work page doi:10.1137/1.9781611974065.ch1 2015

[15] [15]

Dolean, P

V. Dolean, P. Jolivet, P.-H. Tournier, and S. Operto , Iterative frequency-domain seismic wave solvers based on multi-level domain-decomposition preconditioners , in 82 ^ th Annual EAGE Meeting (Amsterdam), 2020. arXiv:2004.06309

work page arXiv 2020

[16] [16]

El Bouajaji , V

M. El Bouajaji , V. Dolean, M. J. Gander, and S. Lanteri , Optimized S chwarz methods for the time-harmonic M axwell equations with damping , SIAM J. Sci. Comput., 34 (2012), pp. A2048--A2071

work page 2012

[17] [17]

M. J. Gander, L. Halpern, and F. Magoul \`e s , An optimized S chwarz method with two-sided R obin transmission conditions for the H elmholtz equation , Internat. J. Numer. Methods Fluids, 55 (2007), pp. 163--175

work page 2007

[18] [18]

M. J. Gander and A. Tonnoir , Analysis of schwarz methods for convected helmholtz-like equations , SIAM Journal on Scientific Computing, 46 (2024), pp. A1--A22

work page 2024

[19] [19]

S. Gong, I. G. Graham, and E. A. Spence , Domain decomposition preconditioners for high-order discretizations of the heterogeneous H elmholtz equation , IMA J. Numer. Anal., (2020). draa080

work page 2020

[20] [20]

I. G. Graham, E. A. Spence, and E. Vainikko , Recent results on domain decomposition preconditioning for the high-frequency helmholtz equation using absorption , in Modern Solvers for Helmholtz Problems, D. Lahaye, J. Tang, and K. Vuik, eds., Geosystems Mathematics, Birkh \"a user, Cham, 2017, pp. 3--26

work page 2017

[21] [21]

I. G. Graham, E. A. Spence, and J. Zou , Domain decomposition with local impedance conditions for the H elmholtz equation , SIAM J. Numer. Anal., 58 (2020), pp. 2515--2543

work page 2020

[22] [22]

Haddar, P

H. Haddar, P. Joly, and H.-M. Nguyen , Generalized impedance boundary conditions for scattering problems from strongly absorbing obstacles: The case of maxwell's equations , Mathematical Models and Methods in Applied Sciences, 18 (2008), pp. 1787--1827

work page 2008

[23] [23]

W. F. Hall and A. V. Kabakian , A sequence of absorbing boundary conditions for maxwell’s equations , Journal of Computational Physics, 194 (2004), pp. 140--155

work page 2004

[24] [24]

Hecht , New development in freefem++ , J

F. Hecht , New development in freefem++ , J. Numer. Math., 20 (2012), pp. 251--265, https://freefem.org/

work page 2012

[25] [25]

Mattesi, M

V. Mattesi, M. Darbas, and C. Geuzaine , A high-order absorbing boundary condition for 2D time-harmonic elastodynamic scattering problems , Comput. Math. Appl., 77 (2019), pp. 1703--1721

work page 2019

[26] [26]

Royer, C

A. Royer, C. Geuzaine, E. B \'e chet, and A. Modave , A non-overlapping domain decomposition method with perfectly matched layer transmission conditions for the helmholtz equation , Computer Methods in Applied Mechanics and Engineering, 395 (2022), p. 115006

work page 2022

[27] [27]

Tournier, P

P.-H. Tournier, P. Jolivet, and F. Nataf , FFDDM : Freefem domain decomposition method . https://doc.freefem.org/documentation/ffddm/index.html , 2019

work page 2019

[28] [28]

write newline

" write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION not #0 #1 if FUNCTION and 'skip pop #0 if FUNCTION or pop #1 'skip if FUNCTION new.block.checka empty 'skip 'new.block if FUNCTION field.or.null duplicate empty pop "" 'skip ...

work page