Backward Euler method for the Equations of Motion Arising in Oldroyd Fluids of Order One with Nonsmooth Initial Data

TL;DR

This paper analyzes a backward Euler numerical scheme for 2D Oldroyd fluids of order one, providing uniform-in-time error estimates even with nonsmooth initial data, enhancing the understanding of numerical stability and accuracy.

Contribution

It introduces a uniform-in-time error analysis for the backward Euler method applied to Oldroyd fluid equations with nonsmooth initial conditions, which was not previously established.

Findings

Discrete solution estimates are uniformly bounded in time.

Optimal L2-norm error estimates are derived for nonsmooth initial data.

Results depend on a uniqueness condition for stability.

Abstract

In this paper, a backward Euler method 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 a priori error estimate in L2-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.