We study the approximation by means of an iterative method towards strong (and more regular) solutions for incompressible Navier-Stokes equations with mass diffusion. In addition, some convergence rates for the error between the approximation and the exact solution will be given, for weak, strong and more regular norms.

Finite element Galerkin method is applied to equations of motion arising in the Kelvin–Voigt model of viscoelastic fluids for spatial discretization. Some new a priori bounds which reflect the exponential decay property are obtained for the exact solution. For optimal L∞(L2) estimate in the velocity, a new auxiliary operator which is based on a modification of the Stokes operator is introduced and analyzed. Finally, optimal error bounds for the velocity in L∞(L2) as well as in L∞(H)‐norms and the pressure in L∞(L2)‐norm are derived which again preserves the exponential decay property. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

In this paper, a linearized backward Euler method is discussed for the equations of motion arising in the Oldroyd model of viscoelastic fluids. Some new a priori bounds are obtained for the solution under realistically assumed conditions on the data. Further, the exponential decay properties for the exact as well as the discrete solutions are established. Finally, a priori error estimates in H 1 and L 2 -norms are derived for the the discrete problem which are valid uniformly for all time t > 0.

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In this article, we analyze a two-level finite element method for the two dimensional timedependent incompressible Navier-Stokes equations with non-smooth initial data. It involves solving the non-linear Navier-Stokes problem on a coarse grid of size H and solving a Stokes problem on a fine grid of size h, h << H. This method gives optimal convergence for velocity in H 1 -norm and for pressure in L 2 -norm. The analysis takes in to account the loss of regularity of the solution at t = 0 of the Navier-Stokes equations.