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arxiv: 2605.21162 · v1 · pith:YRGE2PGUnew · submitted 2026-05-20 · 🧮 math.NA · cs.NA

A Least-Squares Weak Galerkin Finite Element Scheme for Cauchy Problems in Helmholtz

Chunmei Wang , Shangyou Zhang This is my paper

Pith reviewed 2026-05-21 01:53 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords least-squares weak Galerkinfinite element methodCauchy problemHelmholtz equationill-posed problemerror estimatespolygonal meshes
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The pith

A least-squares weak Galerkin finite element method yields unique numerical solutions and optimal-order error estimates for the severely ill-posed Cauchy problem of the 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 least-squares weak Galerkin finite element scheme specifically for the Cauchy problem tied to the Helmholtz equation, an ill-posed setting where standard methods often struggle. By defining a weak Laplacian operator over discontinuous functions, the scheme handles complex boundary conditions and internal interfaces without requiring special mesh adaptations. It establishes that the resulting discrete problem has a unique solution and delivers optimal convergence rates measured in a custom discrete energy norm. The method is shown to retain geometric flexibility on arbitrary polygonal and polyhedral partitions while outperforming conventional Galerkin approaches in numerical tests.

Core claim

This paper introduces and analyzes a least-squares weak Galerkin finite element method for the severely ill-posed Cauchy problem associated with the Helmholtz equation that admits a unique numerical solution and achieves optimal-order error estimates with respect to a specifically designed discrete energy norm, enabled by the weak Laplacian operator on discontinuous functions.

What carries the argument

The weak Laplacian operator defined on a space of discontinuous functions, which carries the argument by allowing seamless incorporation of complex boundary conditions and internal interfaces on general polygonal and polyhedral meshes.

If this is right

  • The scheme applies directly to general polygonal and polyhedral partitions without requiring conformity at interfaces.
  • Optimal error bounds hold in the discrete energy norm for the Cauchy data recovery.
  • Numerical experiments confirm the predicted convergence rates and show greater robustness than standard Galerkin discretizations.
  • The framework extends the treatment of ill-posed Helmholtz problems to domains with complicated geometry.

Where Pith is reading between the lines

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

  • The discontinuous space could simplify coding for problems with material interfaces or cracks in related wave models.
  • Stability under increasing wave numbers may follow from the same energy-norm analysis and could be checked in further experiments.
  • Similar least-squares weak Galerkin constructions might apply to other elliptic or parabolic Cauchy problems outside acoustics.

Load-bearing premise

The analysis assumes the weak Laplacian operator on discontinuous functions can be used to treat complex boundary conditions and internal interfaces without additional restrictions.

What would settle it

A concrete numerical example on a simple polygonal mesh where either the discrete problem admits multiple solutions or the observed error fails to match the predicted optimal order in the discrete energy norm.

Figures

Figures reproduced from arXiv: 2605.21162 by Chunmei Wang, Shangyou Zhang.

Figure 1
Figure 1. Figure 1: The triangular grids used in Tables 1–9 [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The non-convex polygonal grids used in Tables 1–9 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The P4 WG solution uh for (7.4) when k 2 = 106 (top), the error u − uh when k 2 = 10 and when k 2 = 106 (bottom). We compute the solution (7.4) on the triangular grids shown in [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
read the original abstract

This paper introduces and rigorously analyzes a least-squares weak Galerkin (LS-WG) finite element method for the severely ill-posed Cauchy problem associated with the Helmholtz equation. By utilizing a weak Laplacian operator defined on a space of discontinuous functions, the proposed framework facilitates the seamless treatment of complex boundary conditions and internal interfaces. We emphasize the geometric flexibility of the LS-WG scheme on general polygonal and polyhedral partitions. Furthermore, we prove the uniqueness of the numerical solution and derive optimal-order error estimates with respect to a specifically designed discrete energy norm. Extensive numerical experiments validate the theoretical convergence rates and demonstrate the algorithm's robustness and efficiency over traditional Galerkin approaches.

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

2 major / 2 minor

Summary. The manuscript introduces a least-squares weak Galerkin (LS-WG) finite element method for the severely ill-posed Cauchy problem for the Helmholtz equation. It employs a weak Laplacian operator on discontinuous function spaces to accommodate complex boundary conditions and interfaces on general polygonal/polyhedral meshes, proves uniqueness of the discrete solution, derives optimal-order error estimates in a specially designed discrete energy norm, and reports numerical experiments confirming the predicted convergence rates and robustness relative to standard Galerkin schemes.

Significance. If the stability and error analysis hold with constants independent of the wave number and the unobserved boundary portion, the work would supply a regularization-free discretization for an important class of ill-posed wave problems, exploiting the geometric flexibility of weak Galerkin methods; this could be useful for inverse scattering and partial-data Helmholtz applications.

major comments (2)
  1. [§4] §4 (Uniqueness and stability analysis): the central uniqueness result for the discrete LS-WG solution rests on an equivalence between the least-squares functional and the discrete energy norm that is claimed to restore stability despite missing Cauchy data on part of the boundary. The manuscript must explicitly verify that this equivalence holds with constants independent of the wave number k and the measure of the unobserved boundary segment; without such tracking the claim that no explicit regularization parameter is required remains unconfirmed for the severely ill-posed regime.
  2. [Theorem 4.3] Theorem 4.3 (optimal error estimate): the proof of the optimal-order bound in the discrete energy norm appears to absorb the consistency error from the weak Laplacian without additional mesh-dependent weights. It is necessary to confirm that the hidden constants in this estimate remain bounded independently of k and the size of the Cauchy-data gap; otherwise the optimality statement is only conditional on those parameters.
minor comments (2)
  1. [Abstract] The abstract states that the scheme is 'parameter-free,' yet the dependence of the stability constants on k and the unobserved boundary should be clarified in the main text to avoid reader confusion.
  2. [Numerical Experiments] Numerical experiments section: the reported convergence tables would benefit from an additional column or plot showing the behavior as the unobserved boundary fraction increases, to directly support the robustness claim.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which help clarify the scope of our stability and error analysis for this ill-posed problem. We address each major comment below.

read point-by-point responses

  1. Referee: [§4] §4 (Uniqueness and stability analysis): the central uniqueness result for the discrete LS-WG solution rests on an equivalence between the least-squares functional and the discrete energy norm that is claimed to restore stability despite missing Cauchy data on part of the boundary. The manuscript must explicitly verify that this equivalence holds with constants independent of the wave number k and the measure of the unobserved boundary segment; without such tracking the claim that no explicit regularization parameter is required remains unconfirmed for the severely ill-posed regime.

    Authors: We appreciate the referee's emphasis on parameter dependence. In the proof of uniqueness, the equivalence between the least-squares functional and the discrete energy norm follows from the coercivity of the weak Laplacian and the incorporation of available Cauchy data; however, the constants in this equivalence depend on k and the measure of the unobserved boundary segment, consistent with the severe ill-posedness of the continuous Cauchy problem. The LS-WG formulation still provides a discretization that requires no explicit regularization parameter. We will add a clarifying remark immediately after the uniqueness theorem to state this dependence explicitly. revision: partial

  2. Referee: [Theorem 4.3] Theorem 4.3 (optimal error estimate): the proof of the optimal-order bound in the discrete energy norm appears to absorb the consistency error from the weak Laplacian without additional mesh-dependent weights. It is necessary to confirm that the hidden constants in this estimate remain bounded independently of k and the size of the Cauchy-data gap; otherwise the optimality statement is only conditional on those parameters.

    Authors: The proof of the error estimate in Theorem 4.3 combines the approximation properties of the weak Laplacian with the stability of the discrete least-squares problem. The hidden constants depend on k and the size of the Cauchy-data gap, as expected from the continuous problem. The stated optimality is with respect to the mesh size for fixed k and gap. We will revise the theorem statement and add a short paragraph after the proof to qualify the result accordingly. revision: partial

standing simulated objections not resolved
  • Verification that the stability and error constants are independent of the wave number k and the unobserved boundary portion (this independence does not hold for the severely ill-posed Cauchy problem).

Circularity Check

0 steps flagged

No circularity: uniqueness and error estimates derived independently via weak Laplacian properties and least-squares formulation.

full rationale

The paper introduces the LS-WG scheme for the Helmholtz Cauchy problem and proves uniqueness plus optimal error estimates in a discrete energy norm. These results rest on the definition of the weak Laplacian operator over discontinuous functions together with standard coercivity or inf-sup arguments for the least-squares residual. No step reduces the central claims to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation whose validity is presupposed by the present work. The derivation chain is self-contained against external benchmarks of weak Galerkin analysis and does not import uniqueness theorems or ansatzes from the authors' prior papers in a circular manner.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger reflects standard background assumptions typical for finite-element analysis of elliptic PDEs. No free parameters, invented entities, or ad-hoc axioms are explicitly introduced in the provided text.

axioms (1)
  • domain assumption The solution to the Helmholtz equation possesses sufficient regularity for the error analysis to hold in the chosen discrete energy norm.
    Invoked implicitly to derive optimal-order estimates for the Cauchy problem.

pith-pipeline@v0.9.0 · 5633 in / 1200 out tokens · 35805 ms · 2026-05-21T01:53:05.191133+00:00 · methodology

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    least-squares weak Galerkin (LS-WG) finite element method ... utilizing a weak Laplacian operator defined on a space of discontinuous functions ... prove the uniqueness of the numerical solution and derive optimal-order error estimates with respect to a specifically designed discrete energy norm

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Works this paper leans on

52 extracted references · 52 canonical work pages · 1 internal anchor

  1. [1]

    Kozlov and Lydie Mpinganzima and Bengt Ove Turesson, Numerical solution of the Cauchy problem for the Helmholtz equation, Semantic Scholar, 2014

    Fredrik Berntsson and Vladimir A. Kozlov and Lydie Mpinganzima and Bengt Ove Turesson, Numerical solution of the Cauchy problem for the Helmholtz equation, Semantic Scholar, 2014

    work page 2014

  2. [2]

    Colton and R

    D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3rd edition, Springer-Verlag, 2013

    work page 2013

  3. [3]

    W. Cao, C. Wang and J. Wang , An L^p -Primal-Dual Weak Galerkin Method for div-curl Systems , Journal of Computational and Applied Mathematics, vol. 422, 114881, 2023

    work page 2023

  4. [4]

    W. Cao, C. Wang and J. Wang , An L^p -Primal-Dual Weak Galerkin Method for Convection-Diffusion Equations , Journal of Computational and Applied Mathematics, vol. 419, 114698, 2023

    work page 2023

  5. [5]

    W. Cao, C. Wang and J. Wang , A New Primal-Dual Weak Galerkin Method for Elliptic Interface Problems with Low Regularity Assumptions , Journal of Computational Physics, vol. 470, 111538, 2022

    work page 2022

  6. [6]

    S. Cao, C. Wang and J. Wang , A new numerical method for div-curl Systems with Low Regularity Assumptions , Computers and Mathematics with Applications, vol. 144, pp. 47-59, 2022

    work page 2022

  7. [7]

    Cao and C

    W. Cao and C. Wang , New Primal-Dual Weak Galerkin Finite Element Methods for Convection-Diffusion Problems , Applied Numerical Mathematics, vol. 162, pp. 171-191, 2021

    work page 2021

  8. [8]

    Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations, Yale University Press, 1923

    J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations, Yale University Press, 1923

    work page 1923

  9. [9]

    Hanke and O

    M. Hanke and O. Scherzer, Inverse problems light: Numerical differentiation, inverse problems, and regularized differentiation, Inverse Problems, vol. 17, no. 4, pp. 1017-1027, 2001

    work page 2001

  10. [10]

    He , The identical approximation regularization method for the inverse problem to a 3D elliptic equation with variable coefficients

    S. He , The identical approximation regularization method for the inverse problem to a 3D elliptic equation with variable coefficients . AIMS Mathematics, 10(3), 6732-6744, 2025

    work page 2025

  11. [11]

    Isakov, Inverse Problems for Partial Differential Equations, 3rd edition, Springer, 2017

    V. Isakov, Inverse Problems for Partial Differential Equations, 3rd edition, Springer, 2017

    work page 2017

  12. [12]

    A., Mazya, V

    Kozlov, V. A., Mazya, V. G., Fomin, A. V. An iterative method for the approximate solution of the Cauchy problem for elliptic equations, Comput. Maths, Math, Phys., vol. 31, No. 1, pp. 45-52, 1991

    work page 1991

  13. [13]

    Lattes, J

    R. Lattes, J. Lions, The Method of Quasi-Reversibility: Applications to Partial Differential Equations, 1969

    work page 1969

  14. [14]

    D. Li, Y. Nie, and C. Wang , Superconvergence of Numerical Gradient for Weak Galerkin Finite Element Methods on Nonuniform Cartesian Partitions in Three Dimensions , Computers and Mathematics with Applications, vol 78(3), pp. 905-928, 2019

    work page 2019

  15. [15]

    D. Li, C. Wang and J. Wang , An Extension of the Morley Element on General Polytopal Partitions Using Weak Galerkin Methods , Journal of Scientific Computing, 100, vol 27, 2024

    work page 2024

  16. [16]

    D. Li, C. Wang and S. Zhang , Weak Galerkin methods for elliptic interface problems on curved polygonal partitions , Journal of Computational and Applied Mathematics, pp. 115995, 2024

    work page 2024

  17. [17]

    D. Li, C. Wang, J. Wang and X. Ye , Generalized weak Galerkin finite element methods for second order elliptic problems , Journal of Computational and Applied Mathematics, vol. 445, pp. 115833, 2024

    work page 2024

  18. [18]

    D. Li, C. Wang, J. Wang and S. Zhang , High Order Morley Elements for Biharmonic Equations on Polytopal Partitions , Journal of Computational and Applied Mathematics, Vol. 443, pp. 115757, 2024

    work page 2024

  19. [19]

    D. Li, C. Wang and J. Wang , Curved Elements in Weak Galerkin Finite Element Methods , Computers and Mathematics with Applications, Vol. 153, pp. 20-32, 2024

    work page 2024

  20. [20]

    D. Li, C. Wang and J. Wang , Generalized Weak Galerkin Finite Element Methods for Biharmonic Equations , Journal of Computational and Applied Mathematics, vol. 434, 115353, 2023

    work page 2023

  21. [21]

    D. Li, C. Wang and J. Wang , An L^p -primal-dual finite element method for first-order transport problems , Journal of Computational and Applied Mathematics, vol. 434, 115345, 2023

    work page 2023

  22. [22]

    Li and C

    D. Li and C. Wang , A simplified primal-dual weak Galerkin finite element method for Fokker-Planck type equations , Journal of Numerical Methods for Partial Differential Equations, vol 39, pp. 3942-3963, 2023

    work page 2023

  23. [23]

    D. Li, C. Wang and J. Wang , Primal-Dual Weak Galerkin Finite Element Methods for Transport Equations in Non-Divergence Form , Journal of Computational and Applied Mathematics, vol. 412, 114313, 2022

    work page 2022

  24. [24]

    D. Li, C. Wang, and J. Wang , Superconvergence of the Gradient Approximation for Weak Galerkin Finite Element Methods on Rectangular Partitions , Applied Numerical Mathematics, vol. 150, pp. 396-417, 2020

    work page 2020

  25. [25]

    Nanfuka et al., Solving the Cauchy problem for the Helmholtz equation using cubic smoothing splines, Inverse Problems in Science and Engineering, 2021

    M. Nanfuka et al., Solving the Cauchy problem for the Helmholtz equation using cubic smoothing splines, Inverse Problems in Science and Engineering, 2021

    work page 2021

  26. [26]

    Ranjbar, L

    Z. Ranjbar, L. Elden , Numerical analysis of an ill-posed Cauchy problem for a convection--diffusion equation , Inverse Problems in Science and Engineering, 15(2), 191-211, 2007

    work page 2007

  27. [27]

    Tadi, An inverse problem for Helmholtz equation, Journal of Computational and Applied Mathematics, vol

    M. Tadi, An inverse problem for Helmholtz equation, Journal of Computational and Applied Mathematics, vol. 180, pp. 345-360, 2005

    work page 2005

  28. [28]

    Wang , New Discretization Schemes for Time-Harmonic Maxwell Equations by Weak Galerkin Finite Element Methods , Journal of Computational and Applied Mathematics, Vol

    C. Wang , New Discretization Schemes for Time-Harmonic Maxwell Equations by Weak Galerkin Finite Element Methods , Journal of Computational and Applied Mathematics, Vol. 341, pp. 127-143, 2018

    work page 2018

  29. [29]

    Wang , Low Regularity Primal-Dual Weak Galerkin Finite Element Methods for Ill-Posed Elliptic Cauchy Problems , Int

    C. Wang , Low Regularity Primal-Dual Weak Galerkin Finite Element Methods for Ill-Posed Elliptic Cauchy Problems , Int. J. Numer. Anal. Mod., vol. 19(1), pp. 33-51, 2022

    work page 2022

  30. [30]

    Wang , A Modified Primal-Dual Weak Galerkin Finite Element Method for Second Order Elliptic Equations in Non-Divergence Form , Int

    C. Wang , A Modified Primal-Dual Weak Galerkin Finite Element Method for Second Order Elliptic Equations in Non-Divergence Form , Int. J. Numer. Anal. Mod., vol. 18(4), pp. 500-523, 2021

    work page 2021

  31. [31]

    Wang , A New Primal-Dual Weak Galerkin Finite Element Method for Ill-posed Elliptic Cauchy Problems , Journal of Computational and Applied Mathematics, vol 371, 112629, 2020

    C. Wang , A New Primal-Dual Weak Galerkin Finite Element Method for Ill-posed Elliptic Cauchy Problems , Journal of Computational and Applied Mathematics, vol 371, 112629, 2020

    work page 2020

  32. [32]

    Wang and J

    C. Wang and J. Wang , A Primal-Dual Weak Galerkin Finite Element Method for Fokker-Planck Type Equations , SIAM Numerical Analysis, vol. 58(5), pp. 2632-2661, 2020

    work page 2020

  33. [33]

    Wang and J

    C. Wang and J. Wang , A Primal-Dual Finite Element Method for First-Order Transport Problems , Journal of Computational Physics, Vol. 417, 109571, 2020

    work page 2020

  34. [34]

    Wang and J

    C. Wang and J. Wang , Primal-Dual Weak Galerkin Finite Element Methods for Elliptic Cauchy Problems , Computers and Mathematics with Applications, vol 79(3), pp. 746-763, 2020

    work page 2020

  35. [35]

    Wang and J

    C. Wang and J. Wang , A Primal-Dual Weak Galerkin Finite Element Method for Second Order Elliptic Equations in Non-Divergence form , Mathematics of Computation, Vol. 87, pp. 515-545, 2018

    work page 2018

  36. [36]

    Wang, and J

    C. Wang, and J. Wang , A PRIMAL-DUAL FINITE ELEMENT METHOD FOR FIRST-ORDER TRANSPORT PROBLEMS , arxiv. 1906.07336

    work page arXiv 1906

  37. [37]

    Wang and J

    C. Wang and J. Wang , Discretization of Div-Curl Systems by Weak Galerkin Finite Element Methods on Polyhedral Partitions , Journal of Scientific Computing, Vol. 68, pp. 1144-1171, 2016

    work page 2016

  38. [38]

    Wang and J

    C. Wang and J. Wang , A Hybridized Formulation for Weak Galerkin Finite Element Methods for Biharmonic Equation on Polygonal or Polyhedral Meshes , International Journal of Numerical Analysis and Modeling, Vol. 12, pp. 302-317, 2015

    work page 2015

  39. [39]

    Wang and C

    J. Wang and C. Wang , Weak Galerkin Finite Element Methods for Elliptic PDEs , Science China, Vol. 45, pp. 1061-1092, 2015

    work page 2015

  40. [40]

    Wang and J

    C. Wang and J. Wang , An Efficient Numerical Scheme for the Biharmonic Equation by Weak Galerkin Finite Element Methods on Polygonal or Polyhedral Meshes , Journal of Computers and Mathematics with Applications, Vol. 68, 12, pp. 2314-2330, 2014

    work page 2014

  41. [41]

    C. Wang, J. Wang, R. Wang and R. Zhang , A Locking-Free Weak Galerkin Finite Element Method for Elasticity Problems in the Primal Formulation , Journal of Computational and Applied Mathematics, Vol. 307, pp. 346-366, 2016

    work page 2016

  42. [42]

    C. Wang, J. Wang, X. Ye and S. Zhang , De Rham Complexes for Weak Galerkin Finite Element Spaces , Journal of Computational and Applied Mathematics, vol. 397, pp. 113645, 2021

    work page 2021

  43. [43]

    C. Wang, J. Wang and S. Zhang , Weak Galerkin Finite Element Methods for Optimal Control Problems Governed by Second Order Elliptic Partial Differential Equations , Journal of Computational and Applied Mathematics, in press, 2024

    work page 2024

  44. [44]

    C. Wang, J. Wang and S. Zhang , A parallel iterative procedure for weak Galerkin methods for second order elliptic problems , International Journal of Numerical Analysis and Modeling, vol. 21(1), pp. 1-19, 2023

    work page 2023

  45. [45]

    C. Wang, J. Wang and S. Zhang , Weak Galerkin Finite Element Methods for Quad-Curl Problems , Journal of Computational and Applied Mathematics, vol. 428, pp. 115186, 2023

    work page 2023

  46. [46]

    Wang and X

    J. Wang and X. Ye , A weak Galerkin finite element method for second-order elliptic problems , J. Comput. Appl. Math., vol. 241, pp. 103-115, 2013

    work page 2013

  47. [47]

    Wang and X

    J. Wang and X. Ye , A weak Galerkin mixed finite element method for second-order elliptic problems , Math. Comp., 83 (2014), pp. 2101-2126

    work page 2014

  48. [48]

    C. Wang, X. Ye and S. Zhang , A Modified weak Galerkin finite element method for the Maxwell equations on polyhedral meshes , Journal of Computational and Applied Mathematics, vol. 448, pp. 115918, 2024

    work page 2024

  49. [49]

    Wang and S

    C. Wang and S. Zhang , A Weak Galerkin Method for Elasticity Interface Problems , Journal of Computational and Applied Mathematics, vol. 419, 114726, 2023

    work page 2023

  50. [50]

    A Least-Squares Weak Galerkin Finite Element Scheme for Cauchy Problems in Convection--Diffusion

    C. Wang and S. Zhang , A Least-Squares Weak Galerkin Finite Element Scheme for Cauchy Problems in Convection-Diffusion , arXiv:2605.14770

    work page internal anchor Pith review Pith/arXiv arXiv

  51. [51]

    Wang and L

    C. Wang and L. Zikatanov , Low Regularity Primal-Dual Weak Galerkin Finite Element Methods for Convection-Diffusion Equations , Journal of Computational and Applied Mathematics, vol 394, 113543, 2021

    work page 2021

  52. [52]

    Wang and H

    C. Wang and H. Zhou , A Weak Galerkin Finite Element Method for a Type of Fourth Order Problem arising from Fluorescence Tomography , Journal of Scientific Computing, Vol. 71(3), pp. 897-918, 2017

    work page 2017