pith. sign in

arxiv: 2501.16696 · v3 · submitted 2025-01-28 · 🧮 math.NA · cs.NA

Fourier Analysis of Finite Difference Schemes for the Helmholtz Equation in 1D with Dirichlet Conditions: Sharp Estimates and Relative Errors

Martin J. Gander , Hui Zhang , Haiyang Zhou This is my paper

Pith reviewed 2026-05-23 05:24 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Helmholtz equationfinite differenceFourier analysiserror estimatesDirichlet conditionssharp boundsrelative errorwavenumber explicit
0
0 comments X

The pith

Fourier analysis establishes sharp absolute and relative error orders with matching lower bounds for the centered finite difference scheme on the 1D Helmholtz equation.

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

The paper develops a Fourier analysis approach to derive wavenumber-explicit error estimates for the classical centered finite difference discretization of the one-dimensional indefinite Helmholtz equation with Dirichlet boundary conditions. Under the conditions k greater than 20 and k(kh)^2 over σ_k at most 4 over (π minus 2), with σ_k the distance from k to the nearest multiple of π, it proves that the absolute error of the Fourier interpolant of the discrete solution reaches order (kh)^2 over σ_k^2 in the L2 norm and k(kh)^2 over σ_k^2 in the H1 seminorm, with rigorously matching lower bounds of the same orders. The relative error attains order k(kh)^2 over σ_k in both norms when the solution satisfies the derivative scaling with k squared, assuming appropriate scalings on the source term and boundary data. These sharp estimates, including the lower bounds, allow precise quantification of discretization accuracy for wave problems.

Core claim

The paper claims that for the Fourier interpolant of the discrete solution with homogeneous or inhomogeneous Dirichlet conditions, under k>20 and k(kh)^2/σ_k ≤4/(π-2) together with the source scaling sum k^{-p}||f^{(p)}||=O(1) or |g_i|≍k^{-1}, the worst-case attainable convergence order of the absolute error is (kh)^2/σ_k^2 in L2 and k(kh)^2/σ_k^2 in H1 seminorm, with matching lower bounds established in the same orders; the relative error order is k(kh)^2/σ_k in both norms if ||u^{(p)}||_L2 / ||u^{(p-2)}||_L2 ≍ k^2 for p=2,3.

What carries the argument

Fourier analysis applied to the finite difference scheme, which decomposes solutions and errors into Fourier modes to obtain explicit upper and lower bounds in terms of k and σ_k.

If this is right

  • The Fourier analysis yields the same error orders for both homogeneous and inhomogeneous Dirichlet conditions under the stated scalings.
  • The approach serves as a visual tool for evaluating finite difference schemes with source terms, beyond standard dispersion analysis.
  • The theoretical error orders are corroborated by numerical experiments.
  • The lower bounds match the upper bounds rigorously in the stated orders for the absolute and relative errors.

Where Pith is reading between the lines

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

  • The Fourier mode decomposition technique could extend to analyzing other finite difference or finite element schemes for the Helmholtz equation.
  • If the derivative scaling holds for solutions arising from typical sources in applications, the relative error results would directly inform mesh requirements for controlling accuracy at high wavenumbers.
  • The visual evaluation aspect might identify pollution effects in schemes more readily than dispersion relations alone when forcing terms are present.

Load-bearing premise

The assumptions k>20, k(kh)^2/σ_k ≤4/(π-2), and the solution derivative scaling ||u^{(p)}||/||u^{(p-2)}|| ≍ k^2 for p=2,3, together with the source and boundary data scalings, are required to control the constants and obtain the stated orders.

What would settle it

A numerical test with k>20 and h satisfying k(kh)^2/σ_k ≤4/(π-2), source satisfying the sum condition, and solution satisfying the derivative scaling, where the observed L2 error divided by (kh)^2/σ_k^2 either tends to zero or grows without bound as k increases.

Figures

Figures reproduced from arXiv: 2501.16696 by Haiyang Zhou, Hui Zhang, Martin J. Gander.

Figure 1
Figure 1. Figure 1: Admissible mesh size h for well-posedness of (9). Left: Upper bounds hk (19) and h ∗ k (21) so that all smaller h verifies (20) with σk = 1. Right: Existence of h = O(k −1 ) verifying (20) with σk = 1. Proof. If 2 h sin ξh 2 ≥ n + k π or 2 h sin ξh 2 ≤ (n + k − 1)π, then [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Errors in the H1 -semi-norm for f(x) = C(sin(10πx) + sin(20πx) + sin(40πx) + sin(80πx)). experiments. Let f(x) = C(sin(10πx) + sin(20πx) + sin(40πx) + sin(80πx)) with C such that ∥f∥ = 1, and k ∈ {10, 20, 40, 80}π+1. For each k, we use a very fine mesh with N ≈ 8k 2 to get a reference solution, and a sequence of doubly refined meshes starting with N ≈ k to compute the H1 -semi-norm of the errors. The resul… view at source ↗
Figure 3
Figure 3. Figure 3: Symbol errors ψe (44) for the classical (9) & dispersion free (40), (41), (42): h-refinement ξ (roughly < 5 6 k or < 2 3 k). We also see that the scheme (42) modifying both the discrete Laplacian and the source performs the best. The order of max ψe is h 2 , which corroborates Lemma 11. Next, the influence of the wavenumber k is demonstrated, see [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Symbol errors ψ (44) for the classical (9) & dispersion free (40), (41), (42): kh-refinement 10 -1 10 0 9h 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 A rel classical modify k modify " modify both k = 10: + 1 h = 1=d2ke 10 -2 10 -1 10 0 10 1 9h 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 A rel classical modify k modify " modify both k = 20: + 1 h = 1=d2ke 10 -2 10 -1 10 0 10 1 9h 10 -10 10 -8 10 -6 10 -4 10 -2 A rel… view at source ↗
Figure 5
Figure 5. Figure 5: Symbol errors ψrel (44) for the classical (9) & dispersion free (40), (41), (42): kh fixed 27 [PITH_FULL_IMAGE:figures/full_fig_p027_5.png] view at source ↗
read the original abstract

We consider the Dirichlet problem of the indefinite Helmholtz equation in 1D, $u''+k^2u=f$ in $(0,1)$, $u(0)=g_0$, $u(1)=g_1$, with a constant wavenumber $k\in(0,\infty)\backslash\pi\mathbb{N}$ and a source term $f\in H^p_0(0,1)$, $p\ge 4$. We propose an approach based on Fourier analysis to derive wavenumber explicit sharp estimates of absolute and relative errors of \emph{finite difference} methods. Such results have been well known for \emph{finite element} methods (FEM). We use the approach to analyze the classical centered finite difference scheme. For the Fourier interpolants of the discrete solution with homogeneous (or inhomogeneous) Dirichlet conditions, we show rigorously, under the two assumptions $k>20$ and $k(kh)^2/\sigma_k\le4/(\pi-2)$ with $\sigma_k:=\operatorname{dist}(k,\pi\mathbb{N})$, that the worst case attainable convergence order of the absolute error with $\sum_{p=0}^4k^{-p}\|f^{(p)}\|_{L^2}=O(1)$ (or $|g_i|\asymp k^{-1}$) is $(kh)^2/\sigma_k^2$ in the $L^2$-norm and $k(kh)^2/\sigma_k^2$ in the $H^1$-semi-norm, and that of the relative error is $k(kh)^2/\sigma_k$ in both $L^2$- and $H^1$-semi-norms if $\|u^{(p)}\|_{L^2}/\|u^{(p-2)}\|_{L^2}\asymp k^2$ for $p=2,3$. In particular, the lower bounds of these error estimates are established rigorously in the same orders as the upper bounds, which is the main novelty of this work. We show also that the Fourier analysis approach can be used as a convenient visual tool for evaluating finite difference schemes in presence of source terms, which is beyond the scope of dispersion analysis. The results from the theory and visual analysis are corroborated by numerical experiments.

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.

Referee Report

1 major / 2 minor

Summary. The manuscript develops a Fourier analysis method to derive sharp, explicit estimates for the absolute and relative discretization errors of the centered finite difference scheme applied to the one-dimensional indefinite Helmholtz equation with Dirichlet boundary conditions. Under the assumptions k > 20 and k(kh)^2/σ_k ≤ 4/(π-2), with source term scaling ∑ k^{-p} ||f^{(p)}|| = O(1) and boundary data |g_i| ≍ k^{-1}, it proves that the absolute error of the Fourier interpolant of the discrete solution attains the order (kh)^2/σ_k^2 in L^2 and k(kh)^2/σ_k^2 in the H^1 seminorm, with rigorously matching lower bounds. The relative error is shown to be of order k(kh)^2/σ_k in both norms when the solution satisfies the derivative scaling ||u^{(p)}|| / ||u^{(p-2)}|| ≍ k^2 for p=2,3. The approach is also presented as a visual tool for evaluating schemes in the presence of sources.

Significance. If the results hold, this provides the first set of rigorous, matching upper and lower bounds for finite difference errors in the Helmholtz setting, paralleling known results for finite elements. The Fourier analysis offers a practical visual diagnostic tool that goes beyond classical dispersion analysis, and the numerical experiments corroborate the theory. This contributes to the understanding of pollution effects and error behavior in high-wavenumber problems.

major comments (1)
  1. [Abstract] The assumptions k>20, k(kh)^2/σ_k ≤4/(π-2), and ||u^{(p)}||_L2 / ||u^{(p-2)}||_L2 ≍ k^2 (p=2,3) together with the source/boundary scalings are explicitly required for the stated orders (see abstract); while the claims are conditional, the manuscript should include a short discussion or reference showing that these regimes are representative for typical Helmholtz problems, as they control the constants and are load-bearing for the relative-error claim.
minor comments (2)
  1. The definition of σ_k := dist(k, πℕ) appears inline in the abstract; consider stating it at first use for immediate clarity.
  2. In the numerical experiments section, ensure that the chosen test cases explicitly satisfy the derivative scaling assumption to directly corroborate the relative-error bounds.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive suggestion. We address the single major comment below and will incorporate the requested discussion in the revision.

read point-by-point responses

  1. Referee: [Abstract] The assumptions k>20, k(kh)^2/σ_k ≤4/(π-2), and ||u^{(p)}||_L2 / ||u^{(p-2)}||_L2 ≍ k^2 (p=2,3) together with the source/boundary scalings are explicitly required for the stated orders (see abstract); while the claims are conditional, the manuscript should include a short discussion or reference showing that these regimes are representative for typical Helmholtz problems, as they control the constants and are load-bearing for the relative-error claim.

    Authors: We agree that a short discussion of the assumptions would be helpful for readers. The condition k>20 excludes the low-frequency regime where the analysis is not needed, while k(kh)^2/σ_k ≤4/(π-2) ensures the constants remain controlled away from resonances (σ_k>0). The derivative scaling ||u^{(p)}||/||u^{(p-2)}|| ≍ k^2 for p=2,3 follows directly from the Helmholtz equation when the source term satisfies the given O(1) bound and is the standard high-frequency scaling used in pollution-effect studies. We will add a concise paragraph (with references to representative works such as Ihlenburg-Babuška and related 1D analyses) immediately after the abstract to clarify that these regimes are the ones in which the pollution effect and relative-error behavior are typically studied. This addition will be included in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained via direct Fourier analysis

full rationale

The paper derives wavenumber-explicit upper and lower error bounds for the centered finite-difference scheme by applying Fourier analysis directly to the discrete operator and the continuous Helmholtz problem. The bounds are obtained under explicitly stated assumptions (k>20, k(kh)^2/σ_k ≤4/(π-2), source and solution derivative scalings) and are shown to match in order; no step reduces a claimed prediction to a fitted parameter, self-definition, or load-bearing self-citation. The method is presented as an independent extension of known FEM results, with the central claims resting on the internal Fourier estimates rather than external or circular inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on classical Fourier-series representations of solutions in H^p_0 and standard Sobolev embedding properties; the only domain assumptions are that k avoids multiples of pi and that the stated scaling conditions on the solution hold.

axioms (2)
  • standard math Fourier series and Sobolev-space properties for functions in H^p_0(0,1)
    Invoked to represent both continuous and discrete solutions and to bound the error terms.
  • domain assumption k not in pi N so that sigma_k >0 and the continuous problem is well-posed
    Stated explicitly to exclude resonance cases where the problem becomes singular.

pith-pipeline@v0.9.0 · 5968 in / 1420 out tokens · 57537 ms · 2026-05-23T05:24:14.760284+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

53 extracted references · 53 canonical work pages

  1. [1]

    Finite element dispersion analysis for the three-dimensional second-order scalar wave equation.International Journal for Numerical Methods in Engineering, 35(6):1183–1218, 1992

    Najib N Abboud and Peter M Pinsky. Finite element dispersion analysis for the three-dimensional second-order scalar wave equation.International Journal for Numerical Methods in Engineering, 35(6):1183–1218, 1992

    work page 1992

  2. [2]

    Aghamiry, Ali Gholami, Laure Combe, and St´ ephane Operto

    Hossein S. Aghamiry, Ali Gholami, Laure Combe, and St´ ephane Operto. Accurate 3D frequency- domain seismic wave modeling with the wavelength-adaptive 27-point finite-difference stencil: A tool for full waveform inversion.Geophysics, 87(3):R305–R324, 2022

    work page 2022

  3. [3]

    Discrete dispersion relation for hp-version finite element approximation at high wave number.SIAM Journal on Numerical Analysis, 42(2):553–575, 2004

    Mark Ainsworth. Discrete dispersion relation for hp-version finite element approximation at high wave number.SIAM Journal on Numerical Analysis, 42(2):553–575, 2004

    work page 2004

  4. [4]

    Mathematical and Computational Methods in Photonics and Phononics, volume 235

    Habib Ammari, Brian Fitzpatrick, Hyeonbae Kang, Matias Ruiz, Sanghyeon Yu, and Hai Zhang. Mathematical and Computational Methods in Photonics and Phononics, volume 235. American Mathematical Soc., 2018

    work page 2018

  5. [5]

    Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?SIAM Journal on Numerical Analysis, 34(6):2392–2423, 1997

    Ivo M Babuska and Stefan A Sauter. Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?SIAM Journal on Numerical Analysis, 34(6):2392–2423, 1997

    work page 1997

  6. [6]

    Stability analysis of heteroge- neous Helmholtz problems and finite element solution based on propagation media approximation

    H´ el` ene Barucq, Th´ eophile Chaumont-Frelet, and Christian Gout. Stability analysis of heteroge- neous Helmholtz problems and finite element solution based on propagation media approximation. Mathematics of Computation, 86(307):2129–2157, 2017

    work page 2017

  7. [7]

    Th´ eophile Chaumont-Frelet, Dietmar Gallistl, Serge Nicaise, and J´ erˆ ome Tomezyk. Wavenumber- explicit convergence analysis for finite element discretizations of time-harmonic wave propagation problems with perfectly matched layers.Communications in Mathematical Sciences, 20(1):1–52, 2022

    work page 2022

  8. [8]

    Wavenumber explicit convergence analysis for finite element discretizations of general wave propagation problems.IMA Journal of Numerical Analysis, 40(2):1503–1543, 2020

    Th´ eophile Chaumont-Frelet and Serge Nicaise. Wavenumber explicit convergence analysis for finite element discretizations of general wave propagation problems.IMA Journal of Numerical Analysis, 40(2):1503–1543, 2020

    work page 2020

  9. [9]

    Asymptotic dispersion correction in general finite difference schemes for Helmholtz problems.SIAM Journal on Scientific Computing, 46(2):A670– A696, 2024

    Pierre-Henri Cocquet and Martin J Gander. Asymptotic dispersion correction in general finite difference schemes for Helmholtz problems.SIAM Journal on Scientific Computing, 46(2):A670– A696, 2024

    work page 2024

  10. [10]

    Pierre-Henri Cocquet, Martin J Gander, and Xueshuang Xiang. Closed form dispersion corrections including a real shifted wavenumber for finite difference discretizations of 2D constant coefficient Helmholtz problems.SIAM Journal on Scientific Computing, 43(1):A278–A308, 2021. 28

    work page 2021

  11. [11]

    Arnaud Deraemaeker, Ivo Babuˇ ska, and Philippe Bouillard. Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions.International Journal for Numerical Methods in Engineering, 46(4):471–499, 1999

    work page 1999

  12. [12]

    Preasymptotic error analysis of higher order FEM and CIP-FEM for Helmholtz equation with high wave number.SIAM Journal on Numerical Analysis, 53(2):782–804, 2015

    Yu Du and Haijun Wu. Preasymptotic error analysis of higher order FEM and CIP-FEM for Helmholtz equation with high wave number.SIAM Journal on Numerical Analysis, 53(2):782–804, 2015

    work page 2015

  13. [13]

    Pollution and accuracy of solutions of the Helmholtz equation: A novel perspective from the eigenvalues.Journal of Computational and Applied Mathematics, 395:113549, 2021

    Vandana Dwarka and Cornelis Vuik. Pollution and accuracy of solutions of the Helmholtz equation: A novel perspective from the eigenvalues.Journal of Computational and Applied Mathematics, 395:113549, 2021

    work page 2021

  14. [14]

    Multigrid methods for Helmholtz problems: A convergent scheme in 1D using standard components.Direct and Inverse Problems in Wave Propagation and Applications

    Oliver G Ernst and Martin J Gander. Multigrid methods for Helmholtz problems: A convergent scheme in 1D using standard components.Direct and Inverse Problems in Wave Propagation and Applications. De Gruyer, pages 135–186, 2013

    work page 2013

  15. [15]

    American Mathematical Society, 2022

    Lawrence C Evans.Partial Differential Equations, volume 19. American Mathematical Society, 2022

    work page 2022

  16. [16]

    Qiwei Feng, Bin Han, and Michelle Michelle. Sixth-order compact finite difference method for 2D Helmholtz equations with singular sources and reduced pollution effect.Communications in Computational Physics, 34(3):672–712, 2023

    work page 2023

  17. [17]

    Fernandes and Abimael F

    Daniel T. Fernandes and Abimael F. D. Loula. Quasi optimal finite difference method for Helmholtz problem on unstructured grids.International Journal for Numerical Methods in Engineering, 82(10):1244–1281, 2010

    work page 2010

  18. [18]

    Super-localized orthogonal decomposition for high-frequency Helmholtz problems.SIAM Journal on Scientific Computing, 46(4):A2377–A2397, 2021

    Philip Freese, Moritz Hauck, and Daniel Peterseim. Super-localized orthogonal decomposition for high-frequency Helmholtz problems.SIAM Journal on Scientific Computing, 46(4):A2377–A2397, 2021

    work page 2021

  19. [19]

    Compact fourth-order finite difference schemes for Helmholtz equation with high wave numbers.Journal of Computational Mathematics, pages 98–111, 2008

    Yiping Fu. Compact fourth-order finite difference schemes for Helmholtz equation with high wave numbers.Journal of Computational Mathematics, pages 98–111, 2008

    work page 2008

  20. [20]

    Sharp preasymptotic error bounds for the Helmholtzh-FEM

    Jeffrey Galkowski and Euan A Spence. Sharp preasymptotic error bounds for the Helmholtzh-FEM. SIAM Journal on Numerical Analysis, 63(1):1–22, 2025

    work page 2025

  21. [21]

    Stability and finite element error analysis for the Helmholtz equation with variable coefficients.Mathematics of Computation, 89(321):105–138, 2020

    Ivan Graham and Stefan Sauter. Stability and finite element error analysis for the Helmholtz equation with variable coefficients.Mathematics of Computation, 89(321):105–138, 2020

    work page 2020

  22. [22]

    Multi-resolution localized orthogonal decomposition for Helmholtz problems.Multiscale Modeling & Simulation, 20(2):657–684, 2022

    Moritz Hauck and Daniel Peterseim. Multi-resolution localized orthogonal decomposition for Helmholtz problems.Multiscale Modeling & Simulation, 20(2):657–684, 2022

    work page 2022

  23. [23]

    A finite-difference iterative solver of the Helmholtz equation for frequency-domain seismic wave modeling and full-waveform inversion.Geophysics, 86(2):T107– T116, 2021

    Xingguo Huang and Stewart Greenhalgh. A finite-difference iterative solver of the Helmholtz equation for frequency-domain seismic wave modeling and full-waveform inversion.Geophysics, 86(2):T107– T116, 2021

    work page 2021

  24. [24]

    Springer, 1998

    Frank Ihlenburg.Finite Element Analysis of Acoustic Scattering. Springer, 1998

    work page 1998

  25. [25]

    Frank Ihlenburg and Ivo Babuˇ ska. Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation.International Journal for Numerical Methods in Engineering, 38(22):3745–3774, 1995

    work page 1995

  26. [26]

    Finite element solution of the Helmholtz equation with high wave number Part I: theh-version of the FEM.Computers & Mathematics with Applications, 30(9):9–37, 1995

    Frank Ihlenburg and Ivo Babuˇ ska. Finite element solution of the Helmholtz equation with high wave number Part I: theh-version of the FEM.Computers & Mathematics with Applications, 30(9):9–37, 1995

    work page 1995

  27. [27]

    Finite element solution of the Helmholtz equation with high wave number Part II: theh−pversion of the FEM.SIAM Journal on Numerical Analysis, 34(1):315–358, 1997

    Frank Ihlenburg and Ivo Babuska. Finite element solution of the Helmholtz equation with high wave number Part II: theh−pversion of the FEM.SIAM Journal on Numerical Analysis, 34(1):315–358, 1997. 29

    work page 1997

  28. [28]

    Jovanovi´ c and Endre S¨ uli.Analysis of Finite Difference Schemes For Linear Partial Dif- ferential Equations with Generalized Solutions

    Boˇ sko S. Jovanovi´ c and Endre S¨ uli.Analysis of Finite Difference Schemes For Linear Partial Dif- ferential Equations with Generalized Solutions. Springer London, 2014

    work page 2014

  29. [29]

    A sharp relative-error bound for the Helmholtz h-FEM at high frequency.Numerische Mathematik, 150(1):137–178, 2022

    David Lafontaine, Euan A Spence, and Jared Wunsch. A sharp relative-error bound for the Helmholtz h-FEM at high frequency.Numerische Mathematik, 150(1):137–178, 2022

    work page 2022

  30. [30]

    David Lafontaine, Euan A Spence, and Jared Wunsch. Wavenumber-explicit convergence of the hp-FEM for the full-space heterogeneous Helmholtz equation with smooth coefficients.Computers & Mathematics with Applications, 113:59–69, 2022

    work page 2022

  31. [31]

    An optimized CIP-FEM to reduce the pollution errors for the Helmholtz equation on a general unstructured mesh.Journal of Computational Physics, 511:113120, 2024

    Buyang Li, Yonglin Li, and Zongze Yang. An optimized CIP-FEM to reduce the pollution errors for the Helmholtz equation on a general unstructured mesh.Journal of Computational Physics, 511:113120, 2024

    work page 2024

  32. [32]

    FEM and CIP-FEM for Helmholtz equation with high wave number and perfectly matched layer truncation.SIAM Journal on Numerical Analysis, 57(1):96–126, 2019

    Yonglin Li and Haijun Wu. FEM and CIP-FEM for Helmholtz equation with high wave number and perfectly matched layer truncation.SIAM Journal on Numerical Analysis, 57(1):96–126, 2019

    work page 2019

  33. [33]

    Wavenumber explicit convergence of a multi- scale generalized finite element method for heterogeneous Helmholtz problems.SIAM Journal on Numerical Analysis, 61(3):1546–1584, 2023

    Chupeng Ma, Christian Alber, and Robert Scheichl. Wavenumber explicit convergence of a multi- scale generalized finite element method for heterogeneous Helmholtz problems.SIAM Journal on Numerical Analysis, 61(3):1546–1584, 2023

    work page 2023

  34. [34]

    Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation.SIAM Journal on Numerical Analysis, 49(3):1210–1243, 2011

    Jens Markus Melenk and Stefan Sauter. Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation.SIAM Journal on Numerical Analysis, 49(3):1210–1243, 2011

    work page 2011

  35. [35]

    Evan North, Semyon Tsynkov, and Eli Turkel. High-order accurate numerical simulation of monochromatic waves in photonic crystal ring resonators with the help of a non-iterative domain decomposition.Journal of Computational Electronics, 22(1):310–332, 2023

    work page 2023

  36. [36]

    Eliminating the pollution effect in Helmholtz problems by local subscale correction

    Daniel Peterseim. Eliminating the pollution effect in Helmholtz problems by local subscale correction. Mathematics of Computation, 86(305):1005–1036, 2017

    work page 2017

  37. [37]

    Springer, 2016

    Juan Enrique Santos and Patricia Mercedes Gauzellino.Numerical Simulation in Applied Geophysics. Springer, 2016

    work page 2016

  38. [38]

    High-order finite difference methods for the Helmholtz equation.Computer Methods in Applied Mechanics and Engineering, 163(1-4):343–358, 1998

    Ido Singer and Eli Turkel. High-order finite difference methods for the Helmholtz equation.Computer Methods in Applied Mechanics and Engineering, 163(1-4):343–358, 1998

    work page 1998

  39. [39]

    Euan A Spence. A simple proof that the hp-FEM does not suffer from the pollution effect for the constant-coefficient full-space Helmholtz equation.Advances in Computational Mathematics, 49(2):27, 2023

    work page 2023

  40. [40]

    A dispersion minimizing scheme for the 3-D Helmholtz equation based on ray theory.Journal of Computational Physics, 314:618–646, 2016

    Christiaan C Stolk. A dispersion minimizing scheme for the 3-D Helmholtz equation based on ray theory.Journal of Computational Physics, 314:618–646, 2016

    work page 2016

  41. [41]

    Aghamiry, St´ ephane Operto, and Sebastian Riffo

    Pierre-Henri Tournier, Pierre Jolivet, Victorita Dolean, Hossein S. Aghamiry, St´ ephane Operto, and Sebastian Riffo. 3D finite-difference and finite-element frequency-domain wave simulation with mul- tilevel optimized additive Schwarz domain-decomposition preconditioner: A tool for full-waveform inversion of sparse node data sets.Geophysics, 87(5):T381–T...

    work page 2022

  42. [42]

    Group velocity in finite difference schemes.SIAM Review, 24(2):113–136, 1982

    Lloyd N Trefethen. Group velocity in finite difference schemes.SIAM Review, 24(2):113–136, 1982

    work page 1982

  43. [43]

    Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wave number.Journal of Computational Physics, 232(1):272–287, 2013

    Eli Turkel, Dan Gordon, Rachel Gordon, and Semyon Tsynkov. Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wave number.Journal of Computational Physics, 232(1):272–287, 2013

    work page 2013

  44. [44]

    Pollution-free finite difference schemes for non-homogeneous Helmholtz equation.International Journal of Numerical Analysis and Modeling, 11(4), 2014

    Kun Wang and Yau Shu Wong. Pollution-free finite difference schemes for non-homogeneous Helmholtz equation.International Journal of Numerical Analysis and Modeling, 11(4), 2014. 30

    work page 2014

  45. [45]

    High-order compact finite difference methods for solv- ing the high-dimensional Helmholtz equations.Computational Methods in Applied Mathematics, 23(2):491–516, 2023

    Zhi Wang, Yongbin Ge, and Hai-Wei Sun. High-order compact finite difference methods for solv- ing the high-dimensional Helmholtz equations.Computational Methods in Applied Mathematics, 23(2):491–516, 2023

    work page 2023

  46. [46]

    Pre-asymptotic error analysis of CIP-FEM and FEM for the Helmholtz equation with high wave number

    Haijun Wu. Pre-asymptotic error analysis of CIP-FEM and FEM for the Helmholtz equation with high wave number. Part I: linear version.IMA Journal of Numerical Analysis, 34(3):1266–1288, 2014

    work page 2014

  47. [47]

    A dispersion minimizing compact finite difference scheme for the 2D Helmholtz equa- tion.Journal of Computational and Applied Mathematics, 311:497–512, 2017

    Tingting Wu. A dispersion minimizing compact finite difference scheme for the 2D Helmholtz equa- tion.Journal of Computational and Applied Mathematics, 311:497–512, 2017

    work page 2017

  48. [48]

    A phase velocity preserving fourth-order finite dif- ference scheme for the Helmholtz equation with variable wavenumber.Applied Mathematics Letters, 154:109105, 2024

    Tingting Wu, Wenhui Zhang, and Taishan Zeng. A phase velocity preserving fourth-order finite dif- ference scheme for the Helmholtz equation with variable wavenumber.Applied Mathematics Letters, 154:109105, 2024

    work page 2024

  49. [49]

    Fourier beyond dispersion: Wavenumber explicit and precise accuracy of FDMs for the Helmholtz equation.Applied Mathematics Letters, 168:109576, 2025

    Hui Zhang. Fourier beyond dispersion: Wavenumber explicit and precise accuracy of FDMs for the Helmholtz equation.Applied Mathematics Letters, 168:109576, 2025

    work page 2025

  50. [50]

    Dispersion analysis of CIP-FEM for the Helmholtz equation.SIAM Journal on Numerical Analysis, 61(3):1278–1292, 2023

    Yu Zhou and Haijun Wu. Dispersion analysis of CIP-FEM for the Helmholtz equation.SIAM Journal on Numerical Analysis, 61(3):1278–1292, 2023

    work page 2023

  51. [51]

    Preasymptotic error analysis of CIP-FEM and FEM for Helmholtz equation with high wave number

    Lingxue Zhu and Haijun Wu. Preasymptotic error analysis of CIP-FEM and FEM for Helmholtz equation with high wave number. Part II: hp version.SIAM Journal on Numerical Analysis, 51(3):1828–1852, 2013. A Proof of the claim(31) The claim (31) is ϕ(θ) = θ sin2 θ−µ 2 − θ θ2 −µ 2 has no local maximum forθ k < θ < π 2 and 0< µ <1, whereθ k := arcsinµ. The follow...

    work page 2013

  52. [52]

    Note that q 1167 1871 > 7

    work page

  53. [53]

    v(x) > 0 in [

    For 7 10 ≤θ≤ π 2 , we make use of interval arithmetic in Mathematica to showv(θ)>0. The code is given below. a = 7/10; b = Pi/2; n = 100; (*Number of subdivisions*) subintervals = Subdivide[a, b, n]; allIntervalsPositive = True; (*Check each subinterval*) For[i = 1, i <= n, i++, smallInt = Interval[{subintervals[[i]], subintervals[[i + 1]]}]; vSmallInt = ...

    work page