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

TL;DR

This paper analyzes the error decay rates of a fully-discrete finite element method for unsteady p(.,.)-Stokes equations with variable exponent, providing theoretical results and numerical validation for optimal convergence.

Contribution

It introduces a novel error analysis for a fully-discrete finite element scheme applied to unsteady p(.,.)-Stokes equations with variable exponents, including fractional regularity considerations.

Findings

Derived optimal error decay rates for velocity and pressure.

Numerical experiments confirm theoretical error estimates.

Method is stable and effective for p(.,.) ≥ 2.

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 $p(\cdot,\cdot)\ge 2$.