Approximation of time-dependent, viscoelastic fluid
flow: Crank-Nicolson, finite element approximation
Vincent J. Ervin and Norbert Heuer
Numer. Methods Partial Differential Eq. 20 (2), 248-283, 2004.
In this article we analyze a fully discrete approximation
to the time dependent viscoelasticity equations with an Oldroyd B
constitutive equation in $R^d, \, d = 2, 3$.
We use a Crank--Nicolson discretization for the time derivatives.
At each time level a linear system of equations is solved.
To resolve the non-linearities 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, $\Delta t$, sufficiently small and
satisfying $\Delta t \le C h^{\dd/4}$,
existence of the approximate solution is proven.
A priori error estimates for the
approximation in terms of $\Delta t$ and $h$ are also derived.