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arxiv: 2307.14217 · v2 · pith:5RPCROKNnew · submitted 2023-07-26 · 🧮 math.NA · cs.NA

Error estimates for finite element discretizations of the instationary Navier-Stokes equations

Boris Vexler , Jakob Wagner This is my paper

Pith reviewed 2026-05-24 07:53 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords error estimatesNavier-Stokes equationsfinite element discretizationdiscontinuous Galerkinduality argumentGronwall lemmaa priori estimates
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The pith

The L^∞(I;L²(Ω)) error estimate for the Stokes problem extends to the instationary Navier-Stokes equations via error splitting and duality.

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

This paper develops a priori error estimates for a fully discrete approximation of the two-dimensional time-dependent Navier-Stokes equations using discontinuous Galerkin time stepping and inf-sup stable finite elements in space. It extends known best-approximation results from the linear Stokes problem to the nonlinear Navier-Stokes case specifically in the L^∞(I;L²(Ω)) norm. The proof relies on an error splitting technique together with a duality argument that requires stability of both the primal and dual discrete problems. A custom discrete Gronwall lemma is introduced to establish the needed stability bounds. These estimates provide rigorous justification for the accuracy of numerical simulations of unsteady flows.

Core claim

The error between the exact solution of the instationary Navier-Stokes equations and its fully discrete approximation satisfies a best-approximation type bound in the L^∞(I;L²(Ω)) norm, obtained by splitting the error into a discrete part and a continuous part and applying a duality argument to control the discrete part, with stability ensured by a tailored discrete Gronwall lemma.

What carries the argument

An error splitting approach combined with a duality argument for the linearized dual problem, supported by a specially tailored discrete Gronwall lemma to control stability of the discrete solutions.

If this is right

  • The same techniques yield best-approximation error estimates in the L²(I;H¹(Ω)) and L²(I;L²(Ω)) norms.
  • The discrete primal and dual problems remain stable under the assumptions handled by the discrete Gronwall lemma.
  • Error estimates hold for the two-dimensional case with homogeneous Dirichlet boundary conditions.

Where Pith is reading between the lines

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

  • The approach may guide the design of a posteriori error estimators for adaptive computations in fluid dynamics.
  • Extension of the duality argument to three space dimensions would require additional regularity or different stability controls.
  • Similar splitting and duality techniques could apply to other nonlinear evolution equations discretized in the same way.

Load-bearing premise

The discrete solutions must satisfy bounds that allow the specially tailored discrete Gronwall lemma to prevent exponential growth in the stability estimates for the dual problem.

What would settle it

A numerical experiment in which the computed discrete solutions violate the stability assumptions of the discrete Gronwall lemma while the observed error exceeds the best-approximation rate would falsify the applicability of the bound.

read the original abstract

In this work we consider the two dimensional instationary Navier-Stokes equations with homogeneous Dirichlet/no-slip boundary conditions. We show error estimates for the fully discrete problem, where a discontinuous Galerkin method in time and inf-sup stable finite elements in space are used. Recently, best approximation type error estimates for the Stokes problem in the $L^\infty(I;L^2(\Omega))$, $L^2(I;H^1(\Omega))$ and $L^2(I;L^2(\Omega))$ norms have been shown. The main result of the present work extends the error estimate in the $L^\infty(I;L^2(\Omega))$ norm to the Navier-Stokes equations, by pursuing an error splitting approach and an appropriate duality argument. In order to discuss the stability of solutions to the discrete primal and dual equations, a specially tailored discrete Gronwall lemma is presented. The techniques developed towards showing the $L^\infty(I;L^2(\Omega))$ error estimate, also allow us to show best approximation type error estimates in the $L^2(I;H^1(\Omega))$ and $L^2(I;L^2(\Omega))$ norms, which complement this work.

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

1 major / 0 minor

Summary. The paper considers the 2D instationary Navier-Stokes equations with homogeneous Dirichlet boundary conditions and derives error estimates for a fully discrete scheme that combines discontinuous Galerkin time discretization with inf-sup stable finite elements in space. Building on prior best-approximation results for the Stokes problem, the central claim is an extension of the L^∞(I;L²(Ω)) error bound to the nonlinear case via an error-splitting approach combined with a duality argument; a specially tailored discrete Gronwall lemma is introduced to control stability of the discrete primal and dual problems, and the same techniques are used to obtain best-approximation estimates in the L²(I;H¹(Ω)) and L²(I;L²(Ω)) norms.

Significance. If the stability hypotheses of the discrete Gronwall lemma can be verified a priori for the actual discrete Navier-Stokes solution without extra restrictions on data or time step, the result would supply a useful extension of high-order, best-approximation error analysis from the linear Stokes setting to the nonlinear Navier-Stokes equations, strengthening the theoretical foundation for finite-element methods in computational fluid dynamics.

major comments (1)
  1. [Section containing the discrete Gronwall lemma and the duality argument for the L^∞ estimate] The tailored discrete Gronwall lemma (presented to discuss stability of the discrete primal and dual problems) requires hypotheses on the discrete velocity, typically a uniform L^∞(I;L²(Ω)) bound or a smallness condition on the time step or data. These hypotheses are not shown to hold for the discrete Navier-Stokes solution; without such verification the duality argument used for the L^∞(I;L²(Ω)) estimate does not close unconditionally, unlike the Stokes case.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and insightful comments. We address the major comment below.

read point-by-point responses

  1. Referee: [Section containing the discrete Gronwall lemma and the duality argument for the L^∞ estimate] The tailored discrete Gronwall lemma (presented to discuss stability of the discrete primal and dual problems) requires hypotheses on the discrete velocity, typically a uniform L^∞(I;L²(Ω)) bound or a smallness condition on the time step or data. These hypotheses are not shown to hold for the discrete Navier-Stokes solution; without such verification the duality argument used for the L^∞(I;L²(Ω)) estimate does not close unconditionally, unlike the Stokes case.

    Authors: We appreciate the referee highlighting this point. The hypotheses of the tailored discrete Gronwall lemma (a uniform L^∞(I;L²(Ω)) bound on the discrete velocity) are satisfied by the discrete Navier-Stokes solution itself via standard energy estimates that follow from the DG time discretization and the inf-sup stable spatial elements; these estimates hold under the same data and time-step restrictions that guarantee existence of the discrete solution, without introducing new conditions. In the revised manuscript we will insert an explicit verification lemma immediately preceding the duality argument, showing that the required bound follows directly from the discrete energy equality. This closes the argument unconditionally within the paper's setting and makes the extension from the Stokes case fully rigorous. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to Stokes results; central derivation independent

full rationale

The derivation extends L^∞(I;L²) error estimates from the Stokes problem to Navier-Stokes via error splitting plus duality argument, supplemented by a discrete Gronwall lemma for stability. No step reduces a prediction to a fitted parameter by construction, nor does any load-bearing claim collapse to a self-citation chain. The cited Stokes results are treated as prior independent work; the Gronwall lemma is introduced explicitly rather than smuggled in. This is the normal, non-circular outcome for a technical PDE analysis paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard PDE-theory assumptions whose precise statements are not visible in the abstract.

axioms (2)
  • domain assumption The continuous Navier-Stokes problem admits a solution with sufficient regularity for the error analysis to hold
    Error estimates of this type require solution regularity that is typically assumed rather than proved in the abstract.
  • standard math The chosen finite-element spaces are inf-sup stable
    Explicitly stated as part of the spatial discretization.

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