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arxiv: 2605.01617 · v1 · submitted 2026-05-02 · 🧮 math.NA · cs.NA· math.AP

Discontinuity Analysis and Semi-Analytic Spectral Approximation for the Nonlocal Poisson Equation

Thinh Dang , Bacim Alali , Nathan Albin This is my paper

Pith reviewed 2026-05-09 13:45 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.AP
keywords regularitysolutiondiscontinuitiesproblemsourcespectraldiscontinuitykernel
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The pith

Jump discontinuities from the source are inherited by the solution in nonlocal Poisson problems, with kernel regularity controlling higher-order effects, and a smoothing-based semi-analytic spectral method recovers high accuracy by converting to a regular auxiliary problem and adding analytic 1

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

Nonlocal Poisson equations replace local derivatives with integral operators over a horizon, common in peridynamic models for materials. When the source term has jump discontinuities, the solution's regularity depends on the kernel. The analysis shows that jumps in the source are passed directly to the solution. Two main issues arise: if the kernel is singular at the origin, certain derivatives of the solution blow up at the discontinuity location; if the kernel or its derivatives jump at the horizon endpoints, this creates chains of derivative discontinuities at shifted positions. In contrast, very smooth compactly supported kernels avoid these problems and preserve piecewise smooth regularity similar to the source. To compute solutions accurately, the method applies successive smoothing transformations to the problem, constructs explicit correction functions that capture the known discontinuities and singularities, solves the now-regular smoothed problem with a spectral method, and recovers the original solution by adding the corrections back. Numerical tests indicate this reduces Gibbs oscillations and improves convergence rates while keeping the efficiency of spectral approaches.

Core claim

Under general assumptions on compactly supported integrable kernels, we show that jump discontinuities in the source term are inherited by the solution. ... Numerical experiments show substantial gains in accuracy and convergence, demonstrating that the method effectively mitigates the loss of accuracy caused by discontinuities and Gibbs oscillations while retaining the efficiency of spectral methods.

Load-bearing premise

The general assumptions on the kernels (compactly supported and integrable) hold and that the smoothing transformations and correction functions can be explicitly constructed and applied without introducing new errors for the kernels of interest.

read the original abstract

We study a nonlocal Poisson problem with discontinuous source term and analyze how the regularity of the integral kernel determines the discontinuity structure of the corresponding solution. Under general assumptions on compactly supported integrable kernels, we show that jump discontinuities in the source term are inherited by the solution. We then identify two principal mechanisms governing higher-order regularity: singular behavior of the kernel at the origin and jump discontinuities of the kernel, or of its derivatives, at the horizon endpoints. Singularities at the origin lead to blow-up of certain derivatives of the solution at the source discontinuity, while jumps at the horizon generate cascades of derivative discontinuities at translated locations. These phenomena occur for kernels commonly used in peridynamic-type models. By contrast, compactly supported \(C^\infty\) kernels do not generate derivative blow-up or cascading losses of regularity, and in this case the source term and the solution have equivalent piecewise smooth regularity. Motivated by this analysis, we develop a semi-analytic spectral method for the accurate numerical treatment of discontinuous nonlocal problems. The method uses successive smoothing transformations and explicitly constructed correction functions to convert the original problem into an auxiliary problem with improved regularity. A spectral solver is then applied to the smoothed problem, and the approximation to the original solution is recovered by adding back the analytic corrections. Numerical experiments show substantial gains in accuracy and convergence, demonstrating that the method effectively mitigates the loss of accuracy caused by discontinuities and Gibbs oscillations while retaining the efficiency of spectral methods.

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

3 major / 2 minor

Summary. The manuscript analyzes regularity for the nonlocal Poisson equation with discontinuous source term under compactly supported integrable kernels. It proves that source jumps are inherited by the solution, with origin singularities in the kernel causing derivative blow-up at discontinuities and horizon jumps generating cascades of derivative discontinuities at translated points. For C^∞ kernels these effects vanish and piecewise-smooth regularity is preserved. Motivated by the analysis, the authors construct a semi-analytic spectral method that applies successive smoothing transformations and explicit correction functions to produce an auxiliary problem of improved regularity, solves it spectrally, and recovers the original solution by adding back the analytic corrections. Numerical experiments are presented to demonstrate gains in accuracy and convergence rates.

Significance. If the central claims hold, the work supplies a precise regularity theory for a class of nonlocal problems arising in peridynamics and nonlocal mechanics, explaining why standard spectral methods suffer Gibbs phenomena and reduced convergence. The semi-analytic correction approach, when the constructions are explicit and exact, offers a practical route to retain spectral efficiency without sacrificing accuracy on discontinuous data. The combination of kernel-dependent regularity analysis and a targeted numerical remedy is a clear contribution to the numerical analysis of nonlocal PDEs.

major comments (3)
  1. [Abstract and §3] Abstract and §3 (regularity analysis): the inheritance result and the two principal mechanisms (origin blow-up, horizon cascades) are stated for general compactly supported L¹ kernels, yet the manuscript provides no explicit statement of the minimal regularity on the kernel (beyond compact support and integrability) that guarantees the smoothing transformations remain exact and do not re-introduce singularities. Without this, the claim that the auxiliary problem has improved regularity for arbitrary kernels of interest is not fully supported.
  2. [§4] §4 (semi-analytic method): the construction of the correction functions is described as 'explicit,' but no general procedure or closed-form expression is given that works for every integrable kernel; the numerical experiments appear to use only kernels admitting elementary antiderivatives. This leaves open whether quadrature or approximation errors in the corrections offset the reported spectral gains for kernels lacking closed-form primitives.
  3. [§5] §5 (numerical experiments): while accuracy improvements are shown, no a-priori error bound is derived that accounts for the cost or accuracy of building the corrections; the reported convergence rates therefore remain empirical and do not yet confirm that the method 'retains the efficiency of spectral methods' uniformly under the paper’s general kernel assumptions.
minor comments (2)
  1. [§2] Notation for the nonlocal operator and the horizon parameter should be introduced once and used consistently; several instances of ad-hoc symbols appear in the regularity statements.
  2. [§5] Figure captions for the numerical results should include the specific kernel, horizon size, and polynomial degree used in each panel to allow direct reproduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard domain assumptions for nonlocal operators rather than new free parameters or invented entities.

axioms (1)
  • domain assumption Kernels are compactly supported and integrable
    Invoked as the general setting for the discontinuity analysis.

pith-pipeline@v0.9.0 · 5567 in / 1261 out tokens · 31691 ms · 2026-05-09T13:45:57.220243+00:00 · methodology

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