Optimal Error Estimates for Semidiscrete Galerkin approximations to the Equations of Motion Described by Kelvin-Voigt Viscoelastic Fluid Flow Model

TL;DR

This paper develops optimal error estimates for finite element Galerkin approximations of Kelvin-Voigt viscoelastic fluid equations, providing uniform-in-time bounds and numerical validation for the velocity and pressure errors.

Contribution

It introduces new uniform-in-time error bounds for semidiscrete Galerkin methods applied to Kelvin-Voigt fluid models, including velocity and pressure estimates with numerical confirmation.

Findings

Error estimates are valid uniformly in time under certain conditions.

The semidiscrete method admits a global attractor.

Numerical experiments confirm theoretical error bounds.

Abstract

In this paper, the finite element Galerkin method is applied to the equations of motion arising in the Kelvin-Voigt viscoelastic fluid flow model, when the forcing function is in $L^{\infty}(L^2)$. Some a priori estimates for the exact solution, which are valid uniformly in time as $t\mapsto \infty$ and even uniformly in the retardation time $\kappa$ as $\kappa \mapsto 0$, are derived. It is shown that the semidiscrete method admits a global attractor. Further, with the help of a priori bounds and Sobolev-Stokes projection, optimal error estimates for the velocity in $L^{\infty}(L^2)$ and $L^{\infty}(H^1_0)$-norms and for the pressure in $L^{\infty}(L^2)$-norm are established. Since the constants involved in error estimates have an exponential growth in time, therefore, in the last part of the article, under certain uniqueness condition, the error bounds are established which are valid uniformly in time. Finally, some numerical experiments are conducted which confirm our theoretical findings.