Backward Euler method for the equations of motion arising in Oldroyd model of order one with nonsmooth initial data
Provides rigorous numerical analysis for a viscoelastic fluid model, but the contribution is incremental as it extends existing methods to a specific model with nonsmooth data.
The paper develops a backward Euler method with finite element spatial discretization for the 2D Oldroyd model of viscoelastic fluids, proving uniform-in-time boundedness of the discrete solution in Dirichlet norm and optimal L2 error estimates for nonsmooth initial data, validated by numerical experiments.
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.