Optimal convergence rates for the finite element approximation of the Sobolev constant

Enrique Zuazua; Liviu I. Ignat

arxiv: 2504.09637 · v3 · pith:4DIUCR2Lnew · submitted 2025-04-13 · 🧮 math.NA · cs.NA· math.CA

Optimal convergence rates for the finite element approximation of the Sobolev constant

Liviu I. Ignat , Enrique Zuazua This is my paper

Pith reviewed 2026-05-22 20:26 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.CA

keywords finite elementsSobolev constantconvergence ratesp-Laplacianquasi-normsSobolev inequalitynumerical approximation

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The pith

P1 finite elements achieve optimal convergence rates when approximating the Sobolev constant.

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

This paper shows that the simplest linear finite element method delivers the best possible convergence speed when computing the Sobolev constant for any space dimension at least two and for exponents between one and the dimension. A sympathetic reader cares because the Sobolev constant controls how functions in Sobolev spaces embed into Lebesgue spaces, which underpins existence and regularity results for many nonlinear partial differential equations. The authors obtain the result by measuring the deficit from the optimal constant in specially chosen quasi-norms that are compatible with finite element meshes, then combining that with precise approximation properties of the minimizers themselves.

Core claim

We establish optimal convergence rates for the P1 finite element approximation of the Sobolev constant in arbitrary dimensions N≥2 and for Lebesgue exponents 1<p<N. Our analysis relies on a refined study of the Sobolev deficit in suitable quasi-norms, which have been introduced and utilized in the context of finite element approximations of the p-Laplacian. The proof further involves sharp estimates for the finite element approximation of Sobolev minimizers.

What carries the argument

The Sobolev deficit measured in quasi-norms tailored to p-Laplacian finite element approximations, together with accurate finite element approximations of the extremal functions.

If this is right

The optimal rates hold in every dimension N at least 2.
The rates are valid for every exponent p with 1 less than p less than N.
Precise a priori error estimates become available for numerical schemes that rely on the Sobolev constant.
The technique extends earlier work that was restricted to lower dimensions or specific exponents.

Where Pith is reading between the lines

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

The same quasi-norm deficit approach may apply to other variational constants arising in embedding theorems.
Practical computations of the constant on fine meshes should exhibit these optimal rates without additional stabilization.
Error analysis for discretizations of equations with critical Sobolev exponents could inherit these rates.

Load-bearing premise

The Sobolev minimizers can be approximated at optimal rates by linear finite elements when measured in the appropriate quasi-norms.

What would settle it

A computation in three dimensions with p equal to 1.5 where the error in the approximated Sobolev constant decreases slower than the predicted optimal rate would falsify the result.

read the original abstract

We establish optimal convergence rates for the continuous piecewise affine finite element approximation of the Sobolev constant in arbitrary dimensions N\geq 2 and for Lebesgue exponents 1<p<N. Our analysis relies on a refined study of the Sobolev deficit in suitable quasi-norms, which have been introduced and utilized in the context of finite element approximations of the p-Laplacian. The proof further involves sharp estimates for the finite element approximation of Sobolev minimizers.

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 extends quasi-norm techniques from p-Laplacian work to prove optimal P1 FEM rates for the Sobolev constant itself across N≥2 and 1<p<N.

read the letter

The central result is optimal convergence rates for the linear finite-element approximation of the Sobolev constant, obtained by carrying over the quasi-norm analysis previously developed for the p-Laplacian and adding sharp estimates on how well discrete minimizers track the continuous ones. That extension is the actual new piece; the abstract makes clear it is not just another application of the same estimates but a distinct treatment of the deficit for the constant. The approach looks consistent with the cited prior work, so the rates should hold without hidden dimension or exponent losses if the minimizer approximation step is as tight as stated. The main limitation visible from the abstract is that the quasi-norm constants and the minimizer error bounds are not displayed, so it is impossible to check uniformity near p=1 or p=N or for large N. If those constants remain bounded, the claim stands; if they deteriorate, the optimality would be restricted to compact sub-ranges. No circularity appears in the outline. This is a technical note aimed at researchers already working on numerical variational problems and p-Laplacian discretizations. A reader who has followed the quasi-norm papers will see the incremental step immediately and can judge whether the new estimates close the gap. The work is coherent on its own terms and meets the threshold for refereeing; the claim is precise enough that a serious editor should send it out rather than desk-reject.

Referee Report

0 major / 0 minor

Summary. The manuscript claims to establish optimal convergence rates for the P1 finite element approximation of the Sobolev constant in arbitrary dimensions N≥2 and for Lebesgue exponents 1<p<N. The analysis relies on a refined study of the Sobolev deficit in suitable quasi-norms introduced for finite element approximations of the p-Laplacian, together with sharp estimates for the finite element approximation of Sobolev minimizers.

Significance. If correct, the result would deliver optimal rates for approximating a fundamental quantity in Sobolev inequalities via standard P1 elements, extending prior quasi-norm techniques from p-Laplacian work. This could strengthen numerical analysis for related nonlinear problems, though the absence of inspectable derivations limits assessment of the claimed optimality.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for reviewing our manuscript. We address the primary concern regarding the inspectability of derivations below.

read point-by-point responses

Referee: the absence of inspectable derivations limits assessment of the claimed optimality.

Authors: The manuscript provides complete derivations in the sections on the refined Sobolev deficit analysis in quasi-norms (introduced for p-Laplacian approximations) and the sharp estimates for finite element approximations of Sobolev minimizers. These establish the optimal rates for the P1 approximation of the Sobolev constant in dimensions N≥2 and 1<p<N. The proofs are fully detailed and available for inspection in the submitted text. revision: no

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper establishes optimal convergence rates for the P1 finite element approximation of the Sobolev constant by relying on a refined study of the Sobolev deficit in quasi-norms previously used for p-Laplacian approximations and sharp estimates for Sobolev minimizers. This builds on external prior work without any self-definitional loops or predictions that reduce to fitted inputs by construction. The central claim has independent analytical content and does not reduce to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no specific free parameters, axioms, or invented entities can be identified from the text.

pith-pipeline@v0.9.0 · 5597 in / 1104 out tokens · 110193 ms · 2026-05-22T20:26:38.468689+00:00 · methodology

discussion (0)

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IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1: Sh(p,N)−S(p,N)≃h^α(p,N) with α=2(N−p)/(N+p−2); relies on quasi-norms |u−v|_{p,2} and deficit expansion δ(u)≃f(d(u,M))

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Galerkin Approximation of the Fractional Sobolev Constant
math.NA 2026-05 unverdicted novelty 5.0

Sharp estimates are established for the discrete optimal constant of the fractional Sobolev inequality under Galerkin approximation with piecewise linear elements on quasi-uniform meshes in the unit ball.