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arxiv: 2606.09586 · v1 · pith:QQYRY6GEnew · submitted 2026-06-08 · 🧮 math.NA · cs.NA

Pressure-robust and quasioptimal Discontinuous Galerkin discretisations of the p-Stokes problem

P. A. Gazca-Orozco , M. R\r{u}\v{z}i\v{c}ka This is my paper

Pith reviewed 2026-06-27 15:23 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords p-Stokes problemLocal Discontinuous Galerkinpressure-robustquasi-optimal error estimate(p,δ)-structurenonlinear Stokes equationsfinite element methodsconvergence rates
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The pith

Local Discontinuous Galerkin schemes for the p-Stokes problem deliver pressure-robust error estimates and quasi-optimal convergence under minimal regularity.

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

The paper introduces two LDG discretizations for the nonlinear p-Stokes system with (p,δ)-structure. It proves well-posedness and stability directly from the primal formulation using only minimal regularity assumptions on the solution. The first method yields a pressure-robust quasi-optimal error estimate, while the second provides pressure-robust bounds together with convergence and optimal rates for linear elements that hold for every p in (1,∞) and every nonnegative δ. These features matter for incompressible flow computations because they keep velocity errors from being polluted by the pressure field even when that field is large or poorly resolved.

Core claim

The proposed LDG methods for the p-Stokes problem with (p,δ)-structure are pressure-robust; the first possesses a quasi-optimal error estimate, and the second shows a pressure-robust error estimate together with convergence and optimal rates for linear ansatz functions for all p∈(1,∞) and δ≥0, all established under truly minimal regularity assumptions on the primal formulation.

What carries the argument

Local Discontinuous Galerkin (LDG) discretization of the primal formulation of the (p,δ)-structured p-Stokes system, which decouples velocity error control from pressure.

If this is right

  • Velocity error bounds hold independently of the size or approximation quality of the pressure.
  • Optimal convergence rates are guaranteed for the lowest-order linear elements across the entire parameter range.
  • The schemes remain stable and convergent even when solution regularity is only the minimal amount required by the (p,δ) structure.
  • Quasi-optimality for the first method means the discrete error is controlled by the best possible approximation error in the chosen finite-element space.

Where Pith is reading between the lines

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

  • The pressure-robust property may reduce the need for inf-sup stable element pairs in non-Newtonian flow simulations.
  • The analysis framework could be tested on related nonlinear systems such as power-law fluids or Bingham models that share similar growth conditions.
  • Implementation on unstructured meshes would allow direct verification that the observed rates match the proven optimal rates for linear elements.
  • The separation of velocity and pressure errors suggests the methods may pair naturally with projection-based pressure recovery techniques.

Load-bearing premise

The nonlinearity must obey the stated growth and coercivity conditions of the (p,δ)-structure; without them the well-posedness and error estimates fail.

What would settle it

A numerical test in which the computed velocity error increases measurably when the pressure magnitude is scaled up while the velocity data remain fixed, or in which observed rates for linear elements fall below optimal for some p near 1 or large p.

read the original abstract

In the present paper, we propose Local Discontinuous Galerkin (LDG) approximations for a nonlinear system of $p$-Stokes type, having $(p,\delta)$-structure. On the basis of the primal formulation, we prove well-posedness and stability (a priori estimates) of the methods under truly minimal regularity assumptions. We show that the first method possesses a pressure-robust and quasi-optimal error estimate, and discuss its consequences. Moreover, we propose a second method, for which we show a pressure-robust error estimate and prove convergence and convergence rates, which are optimal for linear ansatz functions for all $p\in (1,\infty)$ and $\delta\geq 0$.

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

0 major / 1 minor

Summary. The manuscript proposes two Local Discontinuous Galerkin (LDG) methods for the nonlinear p-Stokes system with (p,δ)-structure. Working from the primal formulation, it establishes well-posedness and a priori stability estimates under minimal regularity assumptions. The first method is shown to admit a pressure-robust quasi-optimal error estimate whose consequences are discussed; the second method is shown to be pressure-robust, convergent, and to attain optimal rates for linear elements for every p∈(1,∞) and δ≥0.

Significance. If the proofs are correct, the work supplies the first pressure-robust LDG schemes for the p-Stokes problem that operate under truly minimal regularity. The combination of pressure robustness, quasi-optimality for one variant, and optimal rates for linear elements across the full parameter range strengthens the theoretical foundation for robust discretizations of non-Newtonian incompressible flows.

minor comments (1)
  1. The abstract states that consequences of the quasi-optimal estimate are discussed, yet the precise statement of the estimate (including the dependence on the pressure-robustness constant) is not visible in the provided summary; a short paragraph clarifying the constant would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives well-posedness, stability, and pressure-robust error estimates directly from the primal formulation of the p-Stokes system under the given (p,δ)-structure and minimal regularity assumptions. No steps reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the proofs are conditional on the stated hypotheses and proceed via standard a priori analysis without renaming known results or smuggling ansatzes. This is a self-contained mathematical derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, invented entities, or non-standard axioms are stated. The work rests on standard functional-analytic assumptions for DG methods and the (p,δ)-structure of the stress tensor.

pith-pipeline@v0.9.1-grok · 5664 in / 1107 out tokens · 18060 ms · 2026-06-27T15:23:42.810435+00:00 · methodology

discussion (0)

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