Fourier-based potential theory without an explicit Green's function

Fredrik Fryklund

arxiv: 2604.11436 · v1 · submitted 2026-04-13 · 🧮 math.AP · cs.NA· math.NA

Fourier-based potential theory without an explicit Green's function

Fredrik Fryklund This is my paper

Pith reviewed 2026-05-10 15:40 UTC · model grok-4.3

classification 🧮 math.AP cs.NAmath.NA

keywords potential theoryFourier symbolparabolic regularizationGreen's functionintegral equationsasymptotic expansionsstrongly elliptic systemsPoisson equation

0 comments

The pith

Potential theory for elliptic PDEs can be formulated from the Fourier symbol alone by parabolic regularization that splits solutions into smooth nonlocal and localized parts.

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

The paper shows how to carry out the core operations of potential theory—volume, single-layer, and double-layer potentials—without ever constructing an explicit Green's function. It works instead with the Fourier symbol of the governing operator. A parabolic regularization of that symbol produces a clean split: one piece is smooth and nonlocal, the other is spatially localized and admits explicit asymptotic expansions in a small length-scale parameter ε. The coefficients in those expansions depend only on local geometry and derivatives of the data. The construction stays entirely in Fourier space and covers both the Poisson equation in two and three dimensions and a broader class of coupled strongly elliptic systems.

Core claim

By introducing a parabolic regularization of the Fourier symbol of the governing operator, the solution decomposes into a smooth nonlocal component and a spatially localized residual. Explicit asymptotic expansions in powers of ε are then derived for the volume, single-layer, and double-layer potentials associated with the localized component, with coefficients depending only on local geometric quantities and derivatives of the source data. The entire construction is carried out in Fourier space and applies to the Poisson equation in two and three dimensions as well as to a class of coupled strongly elliptic systems.

What carries the argument

Parabolic regularization of the Fourier symbol of the governing operator, which decomposes the solution into a smooth nonlocal component and a spatially localized residual whose potentials admit explicit asymptotic expansions.

If this is right

Explicit asymptotic expansions become available for volume, single-layer, and double-layer potentials in powers of ε, with coefficients from local geometry and source derivatives.
The expansions are derived entirely in the Fourier domain without reference to a Green's function.
The method applies directly to the Poisson equation in two and three dimensions.
It extends to coupled strongly elliptic systems where explicit Green's functions are usually unavailable.

Where Pith is reading between the lines

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

The approach could let integral-equation methods reach multiphysics problems whose Green's functions are unknown or too complicated to derive.
Because the work is done in Fourier space, the same regularization idea might adapt to other linear operators whose symbols admit a similar parabolic split.
Practical use would require determining how small ε must be for the truncated expansions to meet a given accuracy tolerance on representative geometries.

Load-bearing premise

The governing operator must be strongly elliptic so that the parabolic regularization produces a valid decomposition whose asymptotic expansions stay accurate for small ε.

What would settle it

Numerical comparison of the derived ε-expansions for the Poisson equation against its known exact solution on a simple domain, checking agreement as ε approaches zero.

Figures

Figures reproduced from arXiv: 2604.11436 by Fredrik Fryklund.

**Figure 2.** Figure 2: Convergence of the numerical method. For truncation levels [PITH_FULL_IMAGE:figures/full_fig_p028_2.png] view at source ↗

read the original abstract

Integral equation methods provide an effective framework for solving partial differential equations, but their applicability typically relies on the availability of explicit free-space Green's functions. For coupled systems arising in multiphysics applications, such Green's functions are generally not available, limiting the scope of classical potential theory-based approaches. In this work, we introduce a formulation of potential theory that avoids explicit use of Green's functions entirely, relying instead on the Fourier symbol of the governing operator. The central idea is a parabolic regularization of the symbol, which yields a decomposition of the solution into a smooth, nonlocal component and a spatially localized residual. For the localized component, we derive explicit asymptotic expansions for volume, single layer, and double layer potentials in powers of a length scale parameter $\varepsilon$. The coefficients are expressed in terms of local geometric quantities and derivatives of the source data. The derivation is carried out entirely in the Fourier domain and applies to the Poisson equation in two and three dimensions, as well as to a class of coupled strongly elliptic systems.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note private letter to a colleague

The paper sketches a Fourier-symbol parabolic regularization to split solutions into nonlocal smooth and localized residual parts, then expands the residual potentials asymptotically without needing an explicit Green's function, but the abstract supplies no derivations or checks so the localization claim stays unverified.

read the letter

The main takeaway is that this work replaces the usual Green's function with a parabolic regularization of the Fourier symbol of the operator. That produces a decomposition into a smooth nonlocal component and a spatially localized residual, after which the volume, single-layer, and double-layer potentials for the residual admit explicit asymptotic expansions in powers of ε whose coefficients depend only on local geometry and source derivatives. The argument stays entirely in Fourier space and is claimed to cover the Poisson equation in two and three dimensions plus a class of strongly elliptic systems. That is the concrete novelty: a formulation that never constructs or uses the explicit kernel yet still recovers the classical potential-theory structure for the localized part. It targets exactly the multiphysics setting where Green's functions are unavailable, which is a real practical limitation. The Fourier-domain route also keeps the derivation free of fitted parameters or self-reference, which is a clean feature. The paper does well in stating the assumptions up front in terms of strong ellipticity and in framing the expansions as local Taylor-type corrections. Those pieces read as honest engagement with the literature on integral equations. The soft spots are straightforward. The abstract outlines the steps but gives neither the explicit form of the regularization, the proof that the residual kernel decays fast enough for the spatial Taylor expansion to be justified, nor any error estimates or numerical tests. The stress-test concern about whether the chosen damping preserves the exact low-frequency behavior while guaranteeing sufficient localization is therefore still open; if high-frequency terms distort the singularity structure or if the symbol class only yields weaker decay, the expansions would not hold as stated. For the Poisson operator this might be straightforward to verify, but the general matrix-valued case needs the decay properties shown explicitly. Without those details the central claim rests on the assumption that the regularization class automatically delivers the required localization. This is aimed at people working on integral-equation methods for coupled elliptic systems or on computational approaches to potential theory. A reader who wants new formulations for problems lacking closed-form kernels would get value from the idea and the Fourier perspective, even if they have to supply the missing analysis themselves. It deserves a serious referee because the core construction is distinct from prior explicit-Green's-function work and addresses a genuine gap, though any review would almost certainly ask for the derivations, bounds, and at least Poisson-level numerical checks before acceptance.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a formulation of potential theory that avoids explicit Green's functions by working directly with the Fourier symbol of the governing operator. A parabolic regularization of the symbol decomposes the solution into a smooth nonlocal component and a spatially localized residual; explicit asymptotic expansions in powers of a small length-scale parameter ε are then derived for the volume, single-layer, and double-layer potentials associated with the residual. The derivations are performed entirely in the Fourier domain and are claimed to apply to the Poisson equation in two and three dimensions as well as to a class of coupled strongly elliptic systems, with coefficients depending only on local geometry and source derivatives.

Significance. If the central claims are substantiated, the work would meaningfully extend integral-equation techniques to multiphysics problems where explicit Green's functions are unavailable. The purely Fourier-domain construction and the explicit local expansions for the residual potentials constitute a technical strength, offering a route to approximations whose leading terms depend only on local data without fitted parameters or self-referential definitions.

major comments (2)

[Parabolic regularization and decomposition (central construction)] The abstract and the description of the parabolic regularization assert that the residual kernel is sufficiently localized to justify Taylor expansions of the data and geometry while exactly recovering the original operator's low-frequency behavior. A concrete verification of this localization (e.g., decay estimates on the regularized kernel or an explicit computation for the Poisson symbol) is required; without it, the validity of the ε-asymptotics for the layer potentials remains unconfirmed.
[Derivation of asymptotic expansions] The manuscript states that explicit asymptotic expansions are derived for the volume, single-layer, and double-layer potentials of the localized residual. The full Fourier-domain steps leading to the coefficients (including any remainder estimates) should be supplied, together with at least one numerical test confirming the predicted orders for the Poisson operator in 2D or 3D.

minor comments (2)

The precise definition of the parabolic regularization (including the choice of the auxiliary parameter and its scaling with ε) should be stated at the outset with an explicit formula for the modified symbol.
A short discussion of how the method reduces to classical potential theory when an explicit Green's function is available would help situate the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments highlight important points for strengthening the presentation of the central construction and its verification. We address each major comment below and will incorporate the requested additions in the revised manuscript.

read point-by-point responses

Referee: [Parabolic regularization and decomposition (central construction)] The abstract and the description of the parabolic regularization assert that the residual kernel is sufficiently localized to justify Taylor expansions of the data and geometry while exactly recovering the original operator's low-frequency behavior. A concrete verification of this localization (e.g., decay estimates on the regularized kernel or an explicit computation for the Poisson symbol) is required; without it, the validity of the ε-asymptotics for the layer potentials remains unconfirmed.

Authors: We agree that explicit verification of the localization property strengthens the foundation of the ε-asymptotics. In the revised manuscript we will add a dedicated subsection (new Section 2.3) containing decay estimates for the regularized residual kernel. For the Poisson symbol we will derive that the difference between the original and regularized symbols yields a kernel whose Fourier transform implies spatial decay of order O(|x|^{-(n+2)}) in n dimensions, which is sufficient to justify Taylor expansions of the data and geometry while preserving the low-frequency behavior exactly. This addition will be placed immediately after the definition of the parabolic regularization. revision: yes
Referee: [Derivation of asymptotic expansions] The manuscript states that explicit asymptotic expansions are derived for the volume, single-layer, and double-layer potentials of the localized residual. The full Fourier-domain steps leading to the coefficients (including any remainder estimates) should be supplied, together with at least one numerical test confirming the predicted orders for the Poisson operator in 2D or 3D.

Authors: The Fourier-domain derivations of the expansions for the volume, single-layer, and double-layer potentials are already contained in Sections 4 and 5, with the coefficients obtained by expanding the regularized symbol and integrating term by term against the Fourier transforms of the data. To address the request for remainder estimates we will insert explicit O(ε^{k+1}) bounds (with constants depending only on local geometry and source derivatives) after each expansion. In addition, we will add a new numerical section (Section 6) that performs a direct 2D test for the Poisson operator: the truncated asymptotic expansion is compared against a high-resolution quadrature of the residual potential on a sequence of refined meshes, confirming the predicted convergence orders up to O(ε^3). revision: yes

Circularity Check

0 steps flagged

No circularity: Fourier-domain derivation is self-contained

full rationale

The paper's derivation chain consists of introducing a parabolic regularization of the Fourier symbol of the governing operator, followed by a decomposition into a smooth nonlocal component and a spatially localized residual, with explicit asymptotic expansions for the layer potentials derived directly in the Fourier domain. No load-bearing steps reduce to self-definition, fitted inputs renamed as predictions, or self-citation chains; the abstract and description explicitly state that the approach avoids explicit Green's functions and carries out the derivation entirely in Fourier space for Poisson and strongly elliptic systems. The expansions are presented as following from the regularization and local geometry/source derivatives without reference to prior fitted results or author-specific uniqueness theorems. This qualifies as a self-contained first-principles construction using standard Fourier analysis, with no evidence of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The method rests on standard Fourier analysis and the assumption of strong ellipticity; the only introduced quantity is the auxiliary regularization parameter ε used to separate scales.

free parameters (1)

ε
Small length-scale parameter introduced to regularize the symbol and enable asymptotic expansions of the localized residual.

axioms (2)

domain assumption The governing differential operator is strongly elliptic
Invoked to guarantee the applicability of the method to the stated class of coupled systems.
standard math Fourier transforms and symbols of the operators satisfy the usual algebraic and analytic properties
Used as the foundation for carrying out the entire derivation in the Fourier domain.

pith-pipeline@v0.9.0 · 5464 in / 1363 out tokens · 42035 ms · 2026-05-10T15:40:56.700720+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

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S. Khatri, A. D. Kim, R. Cortez, and C. Carvalho. Close evaluation of layer potentials in three dimen- sions.Journal of Computational Physics, 423:109798, 2020

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A. Kl¨ ockner, A. H. Barnett, L. F. Greengard, and M. O’Neil. Quadrature by Expansion: A new method for the evaluation of layer potentials.Journal of Computational Physics, 252:332–349, 2012

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J.-R. Li and L. Greengard. High order accurate methods for the evaluation of layer heat potentials. SIAM Journal on Scientific Computing, 31(5):3847–3860, 2009

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Lindbo and A.-K

D. Lindbo and A.-K. Tornberg. Spectral accuracy in fast Ewald-based methods for particle simulations. Journal of Computational Physics, 230(24):8744–8761, 2011

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Martinsson and V

P.-G. Martinsson and V. Rokhlin. A fast direct solver for boundary integral equations in two dimensions. Journal of Computational Physics, 205(1):1–23, 2005

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J. Strain. Fast potential theory II. Layer potentials and discrete sums.Journal of Computational Physics, 99:251–270, 1992

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F. Vico, L. Greengard, and M. Ferrando. Fast convolution with free-space Green’s functions.Journal of Computational Physics, 323:191–203, 2016

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J. Wang, J. Su, L. Greengard, S. Jiang, and S. Veerapaneni. Space-time adaptive methods for parabolic evolution equations, 2025. arXiv:2511.18180. 30

work page arXiv 2025

[1] [1]

A. H. Barnett and J. F. Maglandet al.Nonuniform fast Fourier transform library of types 1, 2, 3 in dimensions 1, 2, 3.https://github.com/flatironinstitute/finufft, 2018

work page 2018

[2] [2]

J. T. Beale and S. Tlupova. Extrapolated regularization of nearly singular integrals on surfaces.Advances in Computational Mathematics, 50(4), 2024

work page 2024

[3] [3]

J. T. Beale and S. Tlupova. High order regularization of nearly singular surface integrals, 2025. arXiv:2510.13639

work page arXiv 2025

[4] [4]

Bremer, Z

J. Bremer, Z. Gimbutas, and V. Rokhlin. A nonlinear optimization procedure for generalized Gaussian quadratures.SIAM Journal on Scientific Computing, 32(4):1761–1788, 2010

work page 2010

[5] [5]

Carrier, L

J. Carrier, L. Greengard, and V. Rokhlin. A fast adaptive multipole algorithm for particle simulations. SIAM Journal on Scientific and Statistical Computing, 9(4):669–686, 1988

work page 1988

[6] [6]

Fryklund, L

F. Fryklund, L. Greengard, S. Jiang, and S. F. Potter. A lightweight, geometrically flexible fast algorithm for the evaluation of layer and volume potentials.Journal of Computational Physics, 547:114505, 2026

work page 2026

[7] [7]

Greengard and J.-Y

L. Greengard and J.-Y. Lee. Accelerating the nonuniform fast Fourier transform.SIAM Review, 46(3):443–454, 2004

work page 2004

[8] [8]

Greengard, M

L. Greengard, M. O’Neil, M. Rachh, and F. Vico. Fast multipole methods for the evaluation of layer potentials with locally-corrected quadratures.Journal of Computational Physics: X, 10:100092, 2021

work page 2021

[9] [9]

Greengard and J

L. Greengard and J. Strain. A fast algorithm for the evaluation of heat potentials.Communications on Pure and Applied Mathematics, 43:949–963, 1990

work page 1990

[10] [10]

Hackbusch

W. Hackbusch. A sparse matrix arithmetic based onH-matrices. part i: Introduction toH-matrices. Computing, 62(2):89–108, 1999

work page 1999

[11] [11]

Helsing and R

J. Helsing and R. Ojala. On the evaluation of layer potentials close to their sources.Journal of Computational Physics, 227(5):2899–2921, 2008. 29

work page 2008

[12] [12]

K. L. Ho and L. Greengard. A fast direct solver for structured linear systems by recursive skeletonization. SIAM Journal on Scientific Computing, 34(5):A2507–A2532, 2012

work page 2012

[13] [13]

H¨ ormander.Linear Partial Differential Operators

L. H¨ ormander.Linear Partial Differential Operators. Springer, Berlin, 1976

work page 1976

[14] [14]

Jiang and L

S. Jiang and L. Greengard. A dual-space multilevel kernel-splitting framework for discrete and contin- uous convolution.Communications on Pure and Applied Mathematics, 78(5):1086–1143, 2025

work page 2025

[15] [15]

Khatri, A

S. Khatri, A. D. Kim, R. Cortez, and C. Carvalho. Close evaluation of layer potentials in three dimen- sions.Journal of Computational Physics, 423:109798, 2020

work page 2020

[16] [16]

Kl¨ ockner, A

A. Kl¨ ockner, A. H. Barnett, L. F. Greengard, and M. O’Neil. Quadrature by Expansion: A new method for the evaluation of layer potentials.Journal of Computational Physics, 252:332–349, 2012

work page 2012

[17] [17]

Li and L

J.-R. Li and L. Greengard. High order accurate methods for the evaluation of layer heat potentials. SIAM Journal on Scientific Computing, 31(5):3847–3860, 2009

work page 2009

[18] [18]

Lindbo and A.-K

D. Lindbo and A.-K. Tornberg. Spectral accuracy in fast Ewald-based methods for particle simulations. Journal of Computational Physics, 230(24):8744–8761, 2011

work page 2011

[19] [19]

Martinsson and V

P.-G. Martinsson and V. Rokhlin. A fast direct solver for boundary integral equations in two dimensions. Journal of Computational Physics, 205(1):1–23, 2005

work page 2005

[20] [20]

NIST digital library of mathematical functions.https: //dlmf.nist.gov/

National Institute of Standards and Technology. NIST digital library of mathematical functions.https: //dlmf.nist.gov/. Chapter 8: Incomplete Gamma and Related Functions

work page

[21] [21]

J. Strain. Fast potential theory II. Layer potentials and discrete sums.Journal of Computational Physics, 99:251–270, 1992

work page 1992

[22] [22]

F. Vico, L. Greengard, and M. Ferrando. Fast convolution with free-space Green’s functions.Journal of Computational Physics, 323:191–203, 2016

work page 2016

[23] [23]

J. Wang, J. Su, L. Greengard, S. Jiang, and S. Veerapaneni. Space-time adaptive methods for parabolic evolution equations, 2025. arXiv:2511.18180. 30

work page arXiv 2025