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arxiv: 2604.07202 · v1 · submitted 2026-04-08 · 🧮 math.NA · cs.NA

Robust preconditioning for an HDG discretization of the time-dependent Stokes equations

Esteban Henr\'iquez , Jeonghun J. Lee , Sander Rhebergen This is my paper

Pith reviewed 2026-05-10 17:50 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords discretizationcondensationpreconditionersrobuststaticstokestheoreticaltime-dependent
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The pith

Parameter-robust preconditioners are derived for the statically condensed hybridizable discontinuous Galerkin discretization of the time-dependent Stokes equations.

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

The paper aims to construct preconditioners for the linear systems that result from statically condensing an HDG discretization of the time-dependent Stokes problem. These preconditioners are designed to remain effective no matter the values of physical parameters like viscosity or time step size, and discretization parameters like mesh size or polynomial degree. A sympathetic reader would care because efficient and reliable solvers are essential for simulating viscous flows in engineering and science, where changing parameters often cause standard methods to slow down dramatically. The work builds on prior analysis to prove uniform well-posedness and then gives conditions on a norm for the face unknowns that ensure the robustness carries over after condensation.

Core claim

We extend the analysis to derive new preconditioners that remain robust with respect to all physical and discretization parameters. The construction relies on first establishing uniform well-posedness of the HDG formulation (before static condensation) through appropriately defined norms. Based on this result, we identify sufficient conditions that a norm on the face space must satisfy to guarantee parameter-robustness of the resulting preconditioner for the statically condensed HDG system. Numerical experiments in two and three dimensions verify our theoretical results.

What carries the argument

The sufficient conditions on the norm defined on the face space, which ensure parameter-robustness of the preconditioner for the statically condensed HDG system, derived from the uniform well-posedness of the pre-condensation HDG formulation.

Load-bearing premise

The HDG formulation before static condensation must be uniformly well-posed in appropriately defined norms, and the chosen norm on the face space must satisfy the sufficient conditions needed for the preconditioner to inherit that robustness.

What would settle it

If numerical experiments show that the number of iterations required by the preconditioned iterative solver grows without bound as the viscosity parameter tends to zero or as the time step size is varied independently, while holding other parameters fixed, the claim of parameter-robustness would be falsified.

read the original abstract

We present parameter-robust preconditioners for linear systems that arise after applying static condensation to a hybridizable discontinuous Galerkin (HDG) discretization of the time-dependent Stokes problem. Building upon the theoretical framework introduced in our previous work [SIAM Journal on Scientific Computing, 47(6):A3212-A3238, 2025], we extend the analysis to derive new preconditioners that remain robust with respect to all physical and discretization parameters. The construction relies on first establishing uniform well-posedness of the HDG formulation (before static condensation) through appropriately defined norms. Based on this result, we identify sufficient conditions that a norm on the face space must satisfy to guarantee parameter-robustness of the resulting preconditioner for the statically condensed HDG system. Numerical experiments in two and three dimensions verify our theoretical results.

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 paper develops parameter-robust preconditioners for the statically condensed linear systems arising from an HDG discretization of the time-dependent Stokes equations. It extends the authors' prior framework by first proving uniform well-posedness of the pre-condensation HDG formulation via suitably chosen norms, then deriving sufficient conditions on a norm for the face space that transfer this robustness to the condensed system. The claims are supported by numerical experiments in two and three dimensions across relevant parameter regimes.

Significance. If the central claims hold, the work advances robust solver technology for HDG methods on time-dependent incompressible flows, addressing a practical need for preconditioners that remain effective under variation of viscosity, time-step size, polynomial degree, and mesh size. The explicit reliance on uniform well-posedness before condensation and the numerical verification across 2D/3D regimes constitute clear strengths; the extension of the cited SIAM J. Sci. Comput. framework without introducing new parameter dependencies is a positive technical feature.

major comments (2)
  1. [Section 3 (analysis of well-posedness and face-norm conditions)] The abstract and outline indicate that uniform well-posedness of the HDG formulation (before condensation) is established via appropriately defined norms, followed by sufficient conditions on the face-space norm. However, the manuscript does not appear to contain an explicit verification that the chosen face norm satisfies those sufficient conditions once the time-dependent term is included; this step is load-bearing for the parameter-robustness claim.
  2. [Section 5 (numerical results)] The numerical experiments are stated to confirm robustness, yet the presentation does not include a direct comparison of iteration counts or condition-number bounds against the theoretical predictions for the full range of physical and discretization parameters (e.g., small viscosity combined with large time steps). This weakens the link between theory and verification.
minor comments (2)
  1. [Section 2] Notation for the face-space norm and the condensed Schur complement should be introduced with explicit reference to the corresponding definitions in the authors' prior SIAM J. Sci. Comput. paper to improve readability.
  2. [Section 2.1] A few typographical inconsistencies appear in the statement of the time-dependent Stokes problem (e.g., placement of the time derivative relative to the viscous term).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We appreciate the positive assessment of the work's significance and address each major comment below, outlining the revisions we will make to strengthen the presentation.

read point-by-point responses

  1. Referee: [Section 3 (analysis of well-posedness and face-norm conditions)] The abstract and outline indicate that uniform well-posedness of the HDG formulation (before condensation) is established via appropriately defined norms, followed by sufficient conditions on the face-space norm. However, the manuscript does not appear to contain an explicit verification that the chosen face norm satisfies those sufficient conditions once the time-dependent term is included; this step is load-bearing for the parameter-robustness claim.

    Authors: We thank the referee for highlighting this point. Section 3 establishes uniform well-posedness of the pre-condensation HDG formulation using suitably chosen norms and then derives sufficient conditions on the face-space norm that transfer robustness to the condensed system. The specific face norm employed is constructed by extending the norm from our prior SIAM J. Sci. Comput. framework to incorporate the time-dependent term, ensuring by design that the conditions hold. To make this verification fully explicit and address the load-bearing nature of the step, we will add a dedicated paragraph or short subsection in the revised Section 3 that directly confirms the chosen norm satisfies all stated sufficient conditions, including the contributions from the time-dependent term. revision: yes

  2. Referee: [Section 5 (numerical results)] The numerical experiments are stated to confirm robustness, yet the presentation does not include a direct comparison of iteration counts or condition-number bounds against the theoretical predictions for the full range of physical and discretization parameters (e.g., small viscosity combined with large time steps). This weakens the link between theory and verification.

    Authors: We agree that a more explicit link between the theoretical predictions and the numerical results would improve the manuscript. Section 5 already reports iteration counts and timings for the preconditioned systems across 2D and 3D test cases, covering wide ranges of viscosity, time-step size, polynomial degree, and mesh size. In the revised version, we will augment Section 5 with additional tables or figures that directly compare the observed iteration counts and estimated condition numbers to the theoretical bounds for the complete parameter space, with particular emphasis on the regime of small viscosity combined with large time steps. This will provide a clearer quantitative verification of the robustness claims. revision: yes

Circularity Check

0 steps flagged

Minor self-citation; central derivation remains independent

full rationale

The paper explicitly builds on the authors' prior SIAM J. Sci. Comput. framework for HDG preconditioning but adds new analysis for the time-dependent Stokes case, including uniform well-posedness via chosen norms and sufficient conditions on the face-space norm to transfer robustness after static condensation. No step reduces a prediction or uniqueness claim to a fitted input, self-referential definition, or unverified self-citation chain. The extension carries over without introducing new parameter dependencies that would force circularity, and numerical verification is external to the derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard finite-element well-posedness theory and the authors' prior HDG analysis; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Uniform well-posedness of the HDG formulation before static condensation holds in appropriately defined norms
    Invoked to identify sufficient conditions on the face-space norm for preconditioner robustness.

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Cited by 1 Pith paper

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

  1. Preconditioning of a hybridizable discontinuous Galerkin method for the coupled Stokes--Darcy system

    math.NA 2026-04 unverdicted novelty 5.0

    Parameter-robust preconditioners are designed and analyzed for the statically condensed HDG discretization of the Stokes-Darcy system by first proving uniform well-posedness and then applying operator-preconditioning ...

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