Finite-element discretization of a two-dimensional grade two fluid

Prof. Vivette Girault

Department of Mathematics, University of Houston, Houston, Texas 77204-3476

We discretize a simplified model of grade two fluid in two-dimensions: Find a velocity vector $\u$ and a scalar pressure $p$ such that
$$-\nu\Delta \u + {\bf curl}(\u -\alpha\Delta\u)\times \u +\nabla p= \f in \Omega$$
$${\rm div} \u = 0 in \Omega$$
$$\u={\bf 0} on \partial\Omega.$$
Here $\Omega$ is a bounded polygonal domain, $\f$ a given exterior force, $\nu>0$ a given viscosity coefficient and $\alpha \ne 0$ a given normal stress modulus coefficient. This problem models a water solution of polymers, which is a non-newtonian fluid. It has at least one solution, if $\f$ belongs to $H({\rm curl};\O)$. Introducing the auxiliary variable
$z = {\rm curl}(\u -\alpha\Delta\u)$,
we write the problem equivalently in the form
$$-\nu\Delta\u + \z\times \u +\nabla p=\f in \Omega$$
$${\nu \over \alpha} z +\u\cdot\nabla z - {\nu \over \alpha}{\rm curl}\u= {\rm curl}\f$$
$${\rm div}\u = 0 in \Omega$$
$$\u={\bf 0} on \partial\Omega.$$
We discretize $\u$ and $p$ in a pair of stable finite-element subspaces of $H^1_0(\O)^2\times L^2(\O)$, such that $\u$ is exactly divergence-free, and $z$ in a finite-element subspace of $H^1(\O)$. We show that, without restriction on the data, the discrete problem has at least one solution that converges to a solution of the original problem. Error estimates are established under adequate assumptions on the data. This is joint work with L. Ridgway Scott.