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arxiv: 2605.21979 · v1 · pith:KIR2OV7Dnew · submitted 2026-05-21 · 🧮 math.NA · cs.NA

Refined convergence structures of the rectangular Raviart-Thomas element

Yifan Yue , Hongtao Chen , Shuo Zhang This is my paper

Pith reviewed 2026-05-22 04:36 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Raviart-Thomas elementLaplace eigenvalue problemsuperclosenesserror expansionrectangular meshmultiple eigenvaluespost-processingfinite element method
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The pith

The lowest-order rectangular Raviart-Thomas element for the Laplace eigenvalue problem exhibits supercloseness to interpolated eigenfunctions, enabling one-order accuracy gains via post-processing and error expansions for both simple and多个t

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

The paper establishes that the lowest-order rectangular Raviart-Thomas element applied to the Laplace eigenvalue problem has a supercloseness property between its discrete eigenfunctions and the corresponding interpolated ones. This property, obtained through integral expansions of interpolation terms on rectangular meshes, permits straightforward post-processing that raises accuracy by up to one order. The authors further derive asymptotic error expansions that cover both simple and multiple eigenvalues, prove that all discrete eigenvalues converge to the true values from above, and supply a rigorous analysis of multiple-eigenvalue behavior on uniform meshes of the square domain. As an additional result they demonstrate equivalence between the rectangular Raviart-Thomas element and the enriched rotated bilinear element.

Core claim

The lowest-order rectangular Raviart-Thomas element possesses a supercloseness property between the discrete eigenfunctions and the interpolated ones, derived from integral expansions of interpolation terms. This property yields error expansions for both simple and multiple eigenvalues that support Richardson extrapolation and prove convergence from above. On uniform rectangular meshes of the square domain the same property gives a rigorous proof of the convergence behavior for multiple eigenvalues. The element is also shown to be equivalent to the enriched rotated bilinear element.

What carries the argument

The integral expansion of interpolation terms on rectangular meshes, which produces the supercloseness between discrete eigenfunctions and their interpolants.

If this is right

  • Post-processing of the discrete eigenfunctions yields up to one additional order of accuracy.
  • Richardson extrapolation becomes available once the error expansions for simple and multiple eigenvalues are known.
  • All eigenvalues of the discrete problem converge to the exact eigenvalues from above.
  • Multiple eigenvalues on uniform meshes of the square domain obey a specific, rigorously proved convergence pattern.
  • The rectangular Raviart-Thomas element can be replaced by the enriched rotated bilinear element without altering the convergence properties.

Where Pith is reading between the lines

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

  • The same integral-expansion technique may produce analogous supercloseness results for other mixed finite-element schemes on rectangular grids.
  • Implementation cost can be reduced by freely switching between the Raviart-Thomas and enriched rotated bilinear elements on rectangles.
  • Extension of the integral-expansion argument to three-dimensional rectangular hexahedral meshes could yield comparable refinements.
  • On general quadrilateral meshes the proofs would require replacement techniques because they exploit rectangular geometry.

Load-bearing premise

The supercloseness and error expansions hold only when the mesh is rectangular, allowing exact integral expansions of the interpolation error.

What would settle it

A numerical experiment on a rectangular mesh in which the computed eigenvalues approach the true values from below would contradict the claimed convergence from above.

read the original abstract

In this work, we fully explore three refined convergence structures of the lowest-order rectangular Raviart-Thomas element in solving the Laplace eigenvalue problem. Firstly, the scheme possesses a property of supercloseness between the discrete eigenfunctions and the interpolated ones, so that post-processing can be easily constructed to improve the accuracy at most by one order. The essentially skillful method is the integral expansion for interpolation terms. Secondly, based on the supercloseness property, we derive the error expansions for not only simple eigenvalues but also multiple eigenvalues, and provide a rigorous proof for them, based on which Richardson extrapolation can be performed. As a byproduct, we prove that all eigenvalues converge from above. Moreover, by utilizing the supercloseness property and Rayleigh quotient analysis, we give a rigorous proof for the convergence behavior for multiple eigenvalues on uniform meshes for the problem on the square domain. Thirdly, the equivalence between the lowest-order rectangular Raviart-Thomas element and the enriched rotated bilinear element is also indicated. At the last of this work, several numerical experiments are designed to demonstrate our theory.

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 examines refined convergence properties of the lowest-order rectangular Raviart-Thomas finite element for the Laplace eigenvalue problem. It establishes supercloseness between discrete and interpolated eigenfunctions via integral expansions, derives error expansions for simple and multiple eigenvalues, proves convergence of eigenvalues from above, provides rigorous analysis for multiple eigenvalues on uniform square meshes, shows equivalence to the enriched rotated bilinear element, and includes numerical experiments.

Significance. This work offers important contributions to superconvergence and error analysis in finite element eigenvalue approximations. The supercloseness property enables post-processing for improved accuracy, and the error expansions support Richardson extrapolation. The proof that eigenvalues converge from above and the equivalence result are valuable theoretical insights. Numerical verification adds credibility to the findings.

major comments (2)
  1. [Abstract] Abstract: The abstract claims derivation of error expansions for multiple eigenvalues with rigorous proofs, but specifies that the rigorous proof for convergence behavior of multiple eigenvalues is only for uniform meshes on the square domain. This creates ambiguity regarding the generality of the results for rectangular meshes, which is central to the paper's title and claims. Please specify the precise scope of each result.
  2. [Section on error expansions for multiple eigenvalues] Section on error expansions for multiple eigenvalues: The use of Rayleigh quotient analysis for multiple eigenvalues appears limited to uniform meshes. If the integral expansion identities used for supercloseness do not hold or require additional assumptions for non-uniform rectangular meshes, the error expansion and post-processing claims may not apply broadly. A concrete example or counterexample on a non-uniform rectangular mesh would strengthen the claim.
minor comments (2)
  1. [Notation] Ensure consistent use of notation for the Raviart-Thomas space and interpolation operators throughout the manuscript.
  2. [Figures] The numerical results figures would benefit from error tables with computed orders to make the convergence rates more evident.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive review of our manuscript on the refined convergence structures of the lowest-order rectangular Raviart-Thomas element. We address each major comment below with clarifications on the scope of our results and indicate the revisions planned for the next version of the paper.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The abstract claims derivation of error expansions for multiple eigenvalues with rigorous proofs, but specifies that the rigorous proof for convergence behavior of multiple eigenvalues is only for uniform meshes on the square domain. This creates ambiguity regarding the generality of the results for rectangular meshes, which is central to the paper's title and claims. Please specify the precise scope of each result.

    Authors: We agree that the abstract should more precisely delineate the scope to avoid ambiguity. The supercloseness property and the associated integral expansion identities are established for general rectangular meshes. Error expansions for both simple and multiple eigenvalues are derived from this supercloseness and hold under the same assumptions. The rigorous proof for the convergence behavior of multiple eigenvalues via Rayleigh quotient analysis is, however, specialized to uniform meshes on the square domain. We will revise the abstract to explicitly state: error expansions and post-processing apply to rectangular meshes, while the detailed Rayleigh-quotient convergence analysis for multiple eigenvalues is proven for uniform square meshes. This revision will align the abstract with the title and the body of the paper. revision: yes

  2. Referee: [Section on error expansions for multiple eigenvalues] Section on error expansions for multiple eigenvalues: The use of Rayleigh quotient analysis for multiple eigenvalues appears limited to uniform meshes. If the integral expansion identities used for supercloseness do not hold or require additional assumptions for non-uniform rectangular meshes, the error expansion and post-processing claims may not apply broadly. A concrete example or counterexample on a non-uniform rectangular mesh would strengthen the claim.

    Authors: The integral expansion identities underlying supercloseness are derived for general rectangular meshes and do not require mesh uniformity. The Rayleigh quotient analysis is applied specifically when proving the convergence behavior of multiple eigenvalues on uniform square meshes. To strengthen the presentation, we will add a numerical example on a non-uniform rectangular mesh demonstrating that the supercloseness property and resulting error expansions continue to hold, thereby supporting the post-processing claims for rectangular meshes. This addition will clarify the applicability of the core results while acknowledging the specialized nature of the Rayleigh-based proof. revision: partial

Circularity Check

0 steps flagged

Derivations rely on independent integral expansions and Rayleigh analysis with no self-referential reduction

full rationale

The paper constructs its central results—supercloseness via integral expansions of interpolation terms, error expansions for simple and multiple eigenvalues, convergence from above, and equivalence to the enriched rotated bilinear element—through direct mathematical proofs on rectangular meshes. These steps use standard finite-element identities and Rayleigh-quotient arguments that are derived within the present work rather than fitted to data or reduced to prior self-citations by construction. The restriction of the multiple-eigenvalue proof to uniform square meshes is an explicit limitation of the analysis, not a hidden circularity. No load-bearing premise collapses to a self-definition or renamed input.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The analysis rests on standard finite-element assumptions for rectangular meshes and solution regularity; no free parameters or invented entities are introduced.

axioms (1)
  • domain assumption Rectangular mesh regularity and sufficient Sobolev regularity of eigenfunctions for interpolation error estimates.
    Standard background assumption invoked for all convergence statements in rectangular Raviart-Thomas analysis.

pith-pipeline@v0.9.0 · 5716 in / 1243 out tokens · 74643 ms · 2026-05-22T04:36:30.596342+00:00 · methodology

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Lean theorems connected to this paper

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

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    the scheme possesses a property of supercloseness between the discrete eigenfunctions and the interpolated ones, so that post-processing can be easily constructed to improve the accuracy at most by one order. [...] we derive the error expansions for not only simple eigenvalues but also multiple eigenvalues

  • IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    by utilizing the supercloseness property and Rayleigh quotient analysis, we give a rigorous proof for the convergence behavior for multiple eigenvalues on uniform meshes for the problem on the square domain

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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

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