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We propose and analyze several finite-element schemes for solving a grade-two
fluid model, with a
tangential boundary condition, in a two-dimensional polygon. The exact
problem is split into a
generalized Stokes problem and a transport equation, in such a way that it
always has a solution
without restriction on the shape of the domain and on the size of the data.
The first scheme uses
divergence-free discrete velocities and a centered discretization of the
transport term, whereas the
other schemes use Hood-Taylor discretizations for the velocity and
pressure, and either a centered or an upwind
discretization of the transport term. One facet of our analysis is
that, without restrictions on the data,
each scheme has a discrete solution and all discrete solutions converge
strongly to solutions of the
exact problem. Furthermore, if the domain is convex and the data satisfy
certain conditions, each
scheme satisfies error inequalities that lead to error estimates.
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