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A Reynolds-semi-robust H(div)-conforming method for unsteady incompressible non-Newtonian flows

Lourenço Beirão da Veiga, Daniele A. Di Pietro, Kirubell B. Haile

TL;DR

The paper tackles the challenge of obtaining velocity error estimates that are robust with respect to Reynolds number and pressure-robust for unsteady non-Newtonian flows governed by the $p$-Navier--Stokes equations. It develops an $\boldsymbol{H}(\operatorname{div})$-conforming Brezzi--Douglas--Marini discretization with a discontinuous Galerkin treatment of the viscous term and reinforced upwind stabilization for convection, enabling regime-dependent, Reynolds-semi-robust error control. A complete velocity-error analysis is provided, including an error equation with time, diffusive, convective, and upwind components, and a Grönwall-based bound that captures pre-asymptotic behavior in convection-dominated regimes. Theoretical results are validated by numerical tests that confirm the predicted convergence rates and demonstrate Reynolds quasi-robustness across a range of rheologies $(r)$ and diffusive parameters $(\nu)$. The approach offers a reliable and accurate framework for simulating unsteady non-Newtonian flows in convection-dominated settings and heterogeneous regimes.

Abstract

In this work, we prove what appear to be the first Reynolds-semi-robust and pressure-robust velocity error estimates for an H(div)-conforming approximation of unsteady incompressible flows of power-law type fluids. The proposed methods 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 convection-dominated flows through regime-dependent estimates of the error contributions. A complete set of numerical results validate the theoretical findings.

A Reynolds-semi-robust H(div)-conforming method for unsteady incompressible non-Newtonian flows

TL;DR

The paper tackles the challenge of obtaining velocity error estimates that are robust with respect to Reynolds number and pressure-robust for unsteady non-Newtonian flows governed by the -Navier--Stokes equations. It develops an -conforming Brezzi--Douglas--Marini discretization with a discontinuous Galerkin treatment of the viscous term and reinforced upwind stabilization for convection, enabling regime-dependent, Reynolds-semi-robust error control. A complete velocity-error analysis is provided, including an error equation with time, diffusive, convective, and upwind components, and a Grönwall-based bound that captures pre-asymptotic behavior in convection-dominated regimes. Theoretical results are validated by numerical tests that confirm the predicted convergence rates and demonstrate Reynolds quasi-robustness across a range of rheologies and diffusive parameters . The approach offers a reliable and accurate framework for simulating unsteady non-Newtonian flows in convection-dominated settings and heterogeneous regimes.

Abstract

In this work, we prove what appear to be the first Reynolds-semi-robust and pressure-robust velocity error estimates for an H(div)-conforming approximation of unsteady incompressible flows of power-law type fluids. The proposed methods 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 convection-dominated flows through regime-dependent estimates of the error contributions. A complete set of numerical results validate the theoretical findings.
Paper Structure (26 sections, 11 theorems, 120 equations, 6 figures, 1 table)

This paper contains 26 sections, 11 theorems, 120 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Setting
  3. Continuous problem
  4. Mesh and inequalities up to a constant
  5. Function spaces
  6. Raviart--Thomas interpolation
  7. Jumps, liftings and discrete gradient
  8. An $\boldsymbol{H}(\operatorname{div})$-conforming scheme
  9. Discrete problem
  10. Summary of the theoretical results
  11. Divergence-free discrete velocity space
  12. Discrete norms and seminorms
  13. Properties of the discrete diffusive form
  14. Stability
  15. Velocity error analysis
  16. ...and 11 more sections

Key Result

Lemma 2

Let $q\in [1,\infty]$, $m\in \{ 0,\ldots,k \}$, and $\boldsymbol{w} \in \boldsymbol{W}^{1,q}_0(\Omega) \cap \boldsymbol{W}^{m+1,q}(\mathcal{T}_{h})$. Then, the following holds for any $T\in \mathcal{T}_{h}$:

Figures (6)

  • Figure 1: Test 1. Error plots for $r = 2$.
  • Figure 2: Test 1. Error plots for $r < 2$.
  • Figure 3: Test 1. Error plots for $r >2$.
  • Figure 4: Test 2. Domain with adopted mesh (left) and streamlines at $T=10$ for $r=2$ (right).
  • Figure 5: Test 2. Streamlines at $t_{\rm F} = 10$.
  • ...and 1 more figures

Theorems & Definitions (27)

  • Remark 1: Choice of model problem
  • Lemma 2: Boundedness and approximation properties of the discrete gradient
  • Remark 3: Time stepping scheme
  • Remark 4: Interpolates of divergence-free functions with zero trace
  • Proposition 5: Norm of the global jump lifting
  • proof
  • Lemma 6: Norm equivalence
  • proof
  • Lemma 7: Boundedness and monotonicity of $a_h$
  • Lemma 8: Existence of a discrete solution and a priori estimate
  • ...and 17 more