Semidiscrete finite element Galerkin approximations to the equations of motion arising in the Oldroyd model

Abstract

In this paper, a semidiscrete finite element Galerkin method for the equations of motion arising in the 2D Oldroyd model of viscoelastic fluids with zero forcing function is analysed. Some new a priori bounds for the exact solutions are derived under realistically assumed conditions on the data. Moreover, the long-time behaviour of the solution is established. By introducing a Stokes-Volterra projection, optimal error bounds for the velocity in the L-infinity(L-2) as well as in the L-infinity(H-1)-norms and for the pressure in the L-infinity(L-2)-norm are derived which are valid uniformly in time t > 0.