On a three step two-grid finite element method for the Oldroyd model of order one

In this work, an optimal error analysis of a three step two-grid method for the equations of motion arising in the 2D Oldroyd model of order one is discussed. The model, which can be thought of as an integral perturbation of Navier-Stokes equations (NSE), represents linear viscoelastic fluid flows. This non-linear model is analyzed here using a three step numerical scheme. In the first step the problem is solved on a coarse grid, and we use this course grid solution to linearize the problem and solve it in second step, on a finer grid. Third step is a correcting step done on the finer grid. Optimal error estimates for the velocity in L infinity(L2) and L infinity(H1)-norms and for the pressure in L infinity(L2)-norm in the semidiscrete case are established. Estimates are shown to be uniform under the uniqueness assumption. Then, based on backward Euler method, a completely discrete scheme is analyzed and optimal a priori error estimates are derived. All the analysis are carried out for the non-smooth initial data. Finally we present some numerical results to validate our theoretical results. These examples show that the three step two-grid method is efficient than solving a nonlinear problem directly, as is expected.