Backward Euler method for the equations of motion arising in Oldroyd model of order one with nonsmooth initial data

TL;DR

This paper analyzes a backward Euler finite element method for 2D Oldroyd viscoelastic fluid equations, providing uniform-in-time error estimates and validating results with numerical experiments.

Contribution

It offers the first uniform-in-time error analysis for the backward Euler method applied to Oldroyd model equations with nonsmooth initial data.

Findings

Discrete solution estimates are uniformly bounded in time.

Optimal a priori error estimates in L2 norm are established.

Numerical results confirm theoretical error bounds.

Abstract

In this paper, a backward Euler method combined with finite element discretization in spatial direction is discussed for the equations of motion arising in the $2D$ Oldroyd model of viscoelastic fluids of order one with the forcing term independent of time or in $L^{\infty}$ in time. It is shown that the estimates of the discrete solution in Dirichlet norm is bounded uniformly in time. Optimal {\it a priori} error estimate in $\textbf{L}^2$-norm is derived for the discrete problem with non-smooth initial data. This estimate is shown to be uniform in time, under the assumption of uniqueness condition. Finally, we present some numerical results to validate our theoretical results.