Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Error analysis for a fully-discrete finite element approximation of the unsteady $p(\cdot,\cdot)$-Stokes equations

Luigi C. Berselli, Alex Kaltenbach, Seungchan Ko

TL;DR

This work develops and analyzes a fully discrete finite element method for the unsteady $p(\cdot,\cdot)$-Stokes equations, using backward Euler in time and conforming, discretely inf-sup stable spaces in space. It leverages variable-exponent and generalized-function frameworks, including $L^{p(\cdot)}$ and $W^{1,p(\cdot)}$, Bochner–Nikolskii and Calderón spaces, to establish a priori error estimates under fractional regularity assumptions on the velocity and pressure; crucial toolchains include the non-linear operator pair $(\mathbf{S},\mathbf{F},\mathbf{F}^*)$ and shifted $N$-functions. The paper proves stability and interpolation results for FE projections, derives fractional interpolation error bounds, and culminates in a main theorem giving rate-decay for velocity and nonlinear stresses with respect to $\tau$ and $h$, plus auxiliary results under a pressure regularity framework. Numerical experiments corroborate the theoretical rates, particularly demonstrating quasi-optimal convergence for $p^{-}\ge 2$, and highlight the practical viability of the approach for non-Newtonian, variable-exponent fluids described by the unsteady $p(\cdot,\cdot)$-Stokes model. The results provide rigorous guidance for accurately simulating smart fluids with spatially and temporally varying rheology in computational contexts.

Abstract

In this paper, we examine a fully-discrete finite element approximation of the unsteady $p(\cdot,\cdot)$-Stokes equations ($i.e.$, $p(\cdot,\cdot)$ is time- and space-dependent), employing a backward Euler step in time and conforming, discretely inf-sup stable finite elements in space. More precisely, we derive error decay rates for the vector-valued velocity field imposing fractional regularity assumptions on the velocity and the kinematic pressure. In addition, we carry out numerical experiments that confirm the optimality of the derived error decay rates in the case $p(\cdot,\cdot)\ge 2$.

Error analysis for a fully-discrete finite element approximation of the unsteady $p(\cdot,\cdot)$-Stokes equations

TL;DR

This work develops and analyzes a fully discrete finite element method for the unsteady -Stokes equations, using backward Euler in time and conforming, discretely inf-sup stable spaces in space. It leverages variable-exponent and generalized-function frameworks, including and , Bochner–Nikolskii and Calderón spaces, to establish a priori error estimates under fractional regularity assumptions on the velocity and pressure; crucial toolchains include the non-linear operator pair and shifted -functions. The paper proves stability and interpolation results for FE projections, derives fractional interpolation error bounds, and culminates in a main theorem giving rate-decay for velocity and nonlinear stresses with respect to and , plus auxiliary results under a pressure regularity framework. Numerical experiments corroborate the theoretical rates, particularly demonstrating quasi-optimal convergence for , and highlight the practical viability of the approach for non-Newtonian, variable-exponent fluids described by the unsteady -Stokes model. The results provide rigorous guidance for accurately simulating smart fluids with spatially and temporally varying rheology in computational contexts.

Abstract

In this paper, we examine a fully-discrete finite element approximation of the unsteady -Stokes equations (, is time- and space-dependent), employing a backward Euler step in time and conforming, discretely inf-sup stable finite elements in space. More precisely, we derive error decay rates for the vector-valued velocity field imposing fractional regularity assumptions on the velocity and the kinematic pressure. In addition, we carry out numerical experiments that confirm the optimality of the derived error decay rates in the case .
Paper Structure (28 sections, 17 theorems, 122 equations, 2 tables)

This paper contains 28 sections, 17 theorems, 122 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Related contributions
  3. Novel contributions of the paper
  4. Preliminaries
  5. Variable Lebesgue and Sobolev spaces, Nikolskiı̆ spaces, and variable Calderón spaces
  6. (Generalized) $N$-functions
  7. Bochner--Nikoskiı̆ spaces
  8. Basic properties of the non-linear operators
  9. Log-Hölder continuity and important related results
  10. The unsteady $p(\cdot,\cdot)$-Stokes equations
  11. Weak formulations
  12. Regularity assumptions
  13. The discrete unsteady $p(\cdot,\cdot)$-Stokes equations
  14. Space discretization
  15. Triangulations
  16. ...and 13 more sections

Key Result

Proposition 2.13

For every $\bfA, \bfB \in \mathbb{R}^{d\times d}$, $r\ge 0$, and a.e. $(t,x)^\top, (t',x')^\top\in Q_T$, we have that

Theorems & Definitions (40)

  • Remark 2.7
  • Proposition 2.13
  • proof
  • Lemma 2.18
  • proof
  • Lemma 2.22: key estimate
  • proof
  • Remark 3.1: equivalent weak formulation
  • Lemma 3.4
  • proof
  • ...and 30 more