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arxiv: 2604.15147 · v1 · submitted 2026-04-16 · 🧮 math.NA · cs.NA

Energy norm error estimates of a hybrid high-order method for the linear parabolic integro-differential equations on general meshes

Achyuta Ranjan Dutta Mohapatra This is my paper

Pith reviewed 2026-05-10 10:01 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords hybrid high-order methodparabolic integro-differential equationserror estimatesCrank-Nicolson schemepolygonal meshesstability analysisBochner norms
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The pith

The equal-order hybrid high-order method combined with Crank-Nicolson time stepping delivers error estimates of order O(τ² + h^{k+1}) for linear parabolic integro-differential equations.

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

This paper presents a numerical scheme for linear parabolic integro-differential equations using the equal-order hybrid high-order method for spatial discretization on general meshes. It analyzes the stability of the semi-discrete scheme and establishes convergence in Bochner norms. A second-order Crank-Nicolson scheme is applied in time, approximating the memory term with composite trapezoidal quadrature, leading to fully discrete error bounds of O(τ² + h^{k+1}) in discrete l² and l∞ norms in H¹ space. These results matter because they provide reliable high-order approximations for equations with memory effects on polygonal meshes without requiring special mesh structures.

Core claim

We design an equal-order HHO spatial discretization for the linear parabolic integro-differential equations, prove stability and convergence for the semi-discrete problem in appropriate norms, then introduce a Crank-Nicolson time discretization with trapezoidal rule for the integral term, analyze stability of the fully discrete scheme, and derive error estimates of order O(τ² + h^{k+1}) in the discrete l²(0,T; H¹(Ω)) and l^∞(0,T; H¹(Ω)) norms, with numerical tests on polygonal meshes confirming the theory.

What carries the argument

The equal-order hybrid high-order (HHO) discretization, which uses local polynomial approximations of degree k on cells and faces, combined with a Crank-Nicolson scheme for time and trapezoidal quadrature for the memory integral.

Load-bearing premise

The exact solution has enough smoothness in space and time so that the error analysis applies, and the trapezoidal rule for the memory term preserves the second-order temporal accuracy.

What would settle it

Numerical simulations on a problem with known exact solution where the observed convergence rate in H1 norm falls below O(τ² + h^{k+1}) for small τ and h would contradict the estimates.

Figures

Figures reproduced from arXiv: 2604.15147 by Achyuta Ranjan Dutta Mohapatra.

Figure 5.1
Figure 5.1. Figure 5.1: Initial triangular (K1 h ), kershaw (K2 h ) and polygonal meshes (K3 h ) (from left to right) [PITH_FULL_IMAGE:figures/full_fig_p032_5_1.png] view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Convergence rates of the error under the the [PITH_FULL_IMAGE:figures/full_fig_p034_5_2.png] view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Convergence rates of the error under the the [PITH_FULL_IMAGE:figures/full_fig_p034_5_3.png] view at source ↗
read the original abstract

We are concerned in designing a suitable numerical scheme based on the equal-order hybrid high-order (HHO) method for the linear parabolic integro-differential equations. The spatial discretization is made using the equal-order HHO method and subsequently we perform the stability analysis of the corresponding semi-discrete scheme. The convergence results are presented in suitably defined Bochner norms for the semi-discrete problem. Then a second-order temporal discretization is implemented on the time domain using a Crank-Nicolson scheme where the memory term is approximated using a composite trapezoidal quadrature rule. The stability of the resultant complete discrete schemes are analyzed followed by derivation of the error estimates of order $\mathcal{O}(\tau^{2}+h^{k+1})$, $k\ge 0$ is the degree of local polynomial approximation, in discrete $l^{2}(0,T;H^{1}(\Omega))$ and $l^{\infty}(0,T;H^{1}(\Omega))$ like norms. Numerical illustrations are performed on some polygonal meshes validating the theoretical estimates.

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.

Circularity Check

0 steps flagged

Standard energy-method derivation with no circular reductions

full rationale

The paper's error analysis for the semi-discrete HHO scheme and fully discrete Crank-Nicolson/trapezoidal scheme follows conventional consistency-plus-stability arguments in Bochner norms. The claimed O(τ² + h^{k+1}) rates are obtained from standard approximation properties of the HHO reconstruction and the quadrature error under the stated regularity assumptions; no step equates a prediction to a fitted input, renames a known result, or reduces the central bound to a self-citation chain. Self-citations (if present) support auxiliary lemmas but are not load-bearing for the main estimates.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard Sobolev-space regularity assumptions and approximation properties of HHO spaces; no free parameters or new entities are introduced.

axioms (1)
  • domain assumption The exact solution belongs to sufficiently high-order Sobolev spaces so that interpolation and quadrature errors can be bounded by the stated powers of h and τ.
    Required for the derivation of O(h^{k+1}) spatial and O(τ²) temporal rates.

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Reference graph

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