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

TL;DR

This paper introduces a novel H(div)-conforming method for unsteady incompressible non-Newtonian flows, providing the first Reynolds-semi-robust and pressure-robust velocity error estimates with validation through numerical experiments.

Contribution

It develops a new discontinuous Galerkin-based approach with stabilization for power-law fluids, offering the first error estimates that are robust to Reynolds number variations.

Findings

Error estimates are validated numerically.

Method performs well in convection-dominated regimes.

Provides regime-dependent error analysis.

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.