A Reynolds-semi-robust H(div)-conforming method for unsteady incompressible power-law flows

Jul 16, 2026

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This work proves Reynolds-semi-robust and pressure-robust velocity error estimates for an H(div)-conforming discretization of unsteady incompressible power-law fluids. The formulation relies on a discontinuous Galerkin approximation of the viscous term and a reinforced upwind-type stabilization for the convective term. Error bounds incorporate pre-asymptotic convergence orders observed in convection-dominated regimes via regime-dependent estimates. A full set of numerical experiments validates the derived theoretical predictions.

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arXiv :2505 .08708v2 [math .NA] 13 Jul 2026 A Reynolds-semi-robust 􀁎(div)-conforming method for unsteady incompressible power-law flows Lourenc¸o Beiro da Veiga 1,2 , Daniele A. Di Pietro3 , and Kirubell B. Haile 1 1Dipartimento di Matematica e Applicazioni, Universit di Milano Bicocca, [lourenco.beirao@unimib.it](lourenco.beirao@unimib.it), [k.haile@campus.unimib.it](k.haile@campus.unimib.it) 2IMATI-PV, CNR, Pavia, Italy 3 IMAG, Univ. Montpellier, CNRS, Montpellier, France, [daniele.di-pietro@umontpellier.fr](daniele.di-pietro@umontpellier.fr) Abstract In this work, we prove what appear to be the first Reynolds-semi-robust and pressure-robust velocity error estimates for an 􀁎(div)-conforming approximation of unsteady incompressible flows of power-law type fluids. The proposed method hinges on a discontinuous Galerkin approximation of the viscous term and a reinforced upwind-type stabilization of the convective term. The derived velocity error estimates account for pre-asymptotic orders of convergence observed in convectiondominated flows through regime-dependent estimates of the error contributions. A complete set of numerical results validate the theoretical findings. Keywords: Navier–Stokes equations, non-Newtonian fluids, 􀁎(div)-conforming methods, Reynoldssemi-robust estimates, pre-asymptotic convergence rates MSC2010 classification: 76A05, 76D05, 65N30, 65N08, 65N12 1 Introduction In this work we prove what appear to be the first Reynolds-semi-robust and pressure-robust error estimates for an 􀁎(div)-conforming finite element approximation of the 􀀿-Navier–Stokes equations. Fluids with nonlinear rheologies are encountered in several fields, ranging from geosciences [1, 47, 71] to chemical engineering [55] and biomechanics [41, 60] . The mathematical study of the corresponding system of equations is considered, e.g., in [5, 13, 36, 59, 64, 69] . Several methods have also been developed for its numerical solution. Finite element methods for creeping flows of non-Newtonian fluids have been considered in [4, 12, 46] . Both finite difference and finite element methods for the full non-Newtonian Navier–Stokes equations are considered in the monograph [24]; see also [25] . Other contributions worth mentioning here include: [37, 57] on finite element methods with implicit power-law-like rheologies; [55, 56] on generalized Newtonian fluids with space variable and concentration-dependent power-law index; [58] as well as the paper series [52–54] concerning various types of discontinuous Galerkin approximations; [48] on transient flows of non-Newtonian fluids. The numerical solution of the 􀀿 -Navier–Stokes problem involves several challenges. The first is obviously related to the non-linearity of the viscous term. The discretization considered here is inspired by discontinuous Galerkin techniques; see, e.g.,[19, 27, 38, 65] . We also borrow ideas from the analysis of the Hybrid High-Order approximations of the 􀀿-Navier-Stokes considered in [15, 22], based on the approach developed in [28, 29] for the scalar case, and from the theoretical developments for the VEM method in [2] . A second challenge is related to robustness in the convection-dominated regime. It is classically known that, in this case, non-dissipative approximations of the convection term can lead to a loss of stability (see for instance [43, 49, 50]) . Remedies to this problem include, but are not limited to: the streamline upwind Petrov–Galerkin method and its variants [8, 16, 40, 72]; the continuous interior penalty method [17, 18]; grad-div stabilizations [26, 67] . A scheme is referred to as Reynolds quasirobust if, assuming sufficient regularity on the solution, it admits velocity error estimates that do not depend on inverse powers of the viscosity coefficient. See [42] for a survey on Reynolds quasi-robust finite element methods for the classical incompressible Navier–Stokes equations. A third challenging aspect, originally pointed

FAQs

What is the main contribution of this work?

It establishes Reynolds-semi-robust and pressure-robust velocity error estimates for an H(div)-conforming approximation of unsteady incompressible power-law flows.

How is the numerical method constructed?

It uses a discontinuous Galerkin approximation for the viscous term and a reinforced upwind-type stabilization for the convective term.

What do the error estimates capture about different flow regimes?

The velocity error bounds include pre-asymptotic convergence orders that depend on convection-dominated behavior and the power-law exponent through regime-dependent contributions.

Jul 16, 2026

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