Approximation of time‐dependent, viscoelastic fluid flow: Crank‐Nicolson, finite element approximation

In this article we analyze a fully discrete approximation to the time dependent viscoelasticity equations with an Oldroyd B constitutive equation in ℝ $^{\acute{d}}$ , $\acute{d}$ = 2, 3. We use a Crank‐Nicolson discretization for the time derivatives. At each time level a linear system of equations is solved. To resolve the nonlinearities we use a three‐step extrapolation for the prediction of the velocity and stress at the new time level. The approximation is stabilized by using a discontinuous Galerkin approximation for the constitutive equation. For the mesh parameter, h , and the temporal step size, Δ t , sufficiently small and satisfying Δ t ≤ Ch $^{\acute{d}/4}$ , existence of the approximate solution is proven. A priori error estimates for the approximation in terms of Δ t and h are also derived. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 248–283, 2004