Two-grid finite element galerkin approximation of equations of motion arising in Oldroyd fluids of order one with non-smooth initial data

We carry out a fully discrete two-grid finite element approximation for the equations of motion arising in the flow of $2D$ Oldroyd fluids. The non-linear parabolic integro-differential equation is solved on a coarse grid. And only a linearized equation is solved on a fine grid, where the linearization is done based on a time dependent Stokes type problem using the coarse grid solution. A first order time discretization scheme based on backward Euler method is then applied. The scheme gives optimal convergence rate for the velocity in $\mathbf{H}^1$-norm and for the pressure in $L^2$-norm. These estimates are shown to be uniform in time under the assumption of uniqueness condition. Numerical results are provided in support of our theoretical findings.