In this paper, the finite element Galerkin method is applied to the equations of motion arising in the Kelvin-Voigt viscoelastic fluid flow model, when the forcing function is in L ∞ (L 2 ). Some a priori estimates for the exact solution, which are valid uniformly in time as t → ∞ and even uniformly in the retardation time κ an κ → 0, are derived. It is shown that the semidiscrete method admits a global attractor. Further, with the help of a priori bounds and Sobolev-Stokes projection, optimal error estimates for the velocity in L ∞ (L 2 ) and L ∞ (H 1 )-norms and for the pressure in L ∞ (L 2 )-norm are established. Since the constants involved in error estimates have an exponential growth in time, therefore, in the last part of the article, under certain uniqueness condition, the error bounds are established which are valid uniformly in time. Finally, some numerical experiments are conducted which confirm our theoretical findings.
In this paper, an error analysis of a three steps two level Galekin finite element method for the two dimensional transient Navier-Stokes equations is discussed. First of all, the problem is discretized in spatial direction by employing finite element method on a coarse mesh T H with mesh size H. Then, in step two, the nonlinear system is linearized around the coarse grid solution, say, u H , which is similar to Newton's type iteration and the resulting linear system is solved on a finer mesh T h with mesh size h. In step three, a correction is obtained through solving a linear problem on the finer mesh and an updated final solution is derived. Optimal error estimates in L ∞ (L 2 )-norm, when h = O(H 2−δ ) and in L ∞ (H 1 )-norm, when h = O(H 4−δ ) for the velocity and in L ∞ (L 2 )-norm, when h = O(H 4−δ ) for the pressure are established for arbitrarily small δ > 0. Further, under uniqueness assumption, these estimates are proved to be valid uniformly in time. Then based on backward Euler method, a completely discrete scheme is analyzed and a priori error estimates are derived. Finally, the paper is concluded with some numerical experiments.
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