Volume 15, Number 4, Winter 2007
REFINED MIXED FINITE ELEMENT
METHOD FOR THE STOKES PROBLEM
HASNA EL SOSSA AND LUC PAQUET
Abstract. We consider the Dirichlet problem for the
Stokes problem in a plane polygonal domain (simply connected)
with a reentrant corner at the origin. In addition to the velocity
field , the tensor field
is introduced as a supplementary unknown leading us to a mixed formulation. Note that
contrarily to [4], that our mixed formulation of the continuous
problem is not hybridized. To discretize the mixed formulation,
given a fixed triangulation, each of the two lines of the tensor
field σ is approximated on each triangle of the given triangulation
by a RaviartThomas vector field of degree 0 and
by a constant vector field. Under suitable refinement conditions on
the regular family of triangulations under consideration ,
we prove that despite to the reentrant corner, that the convergence of the sequence of approximate solutions
to ((σ,p),)
is still of order 1 in the L^{2}norm. Finally, the
discretized mixed formulation is hybridized by introducing a
Lagrange multiplier to relax the continuity of the normal
trace of σ_{h} across interelement edges e_{h} of the triangulation
T_{h}. A formula expressing
in terms of _{h} and σ_{h}
is established. Under some additional assumptions, it is proved that
uniformly in e_{h}, when h → o^{+}.
Without these additional assumptions, the behaviour of the {_{h}}
at the boundary Γ is studied. The jumps of _{h}
as one crosses from one triangle to an adjacent one are studied and
also compared with those of the means of
on each triangle of the triangulation T_{h}.
In the late section about the numerical
implementation, we first derive explicit formulas for σ _{hK}
and _{hK} on each triangle
K of the triangulation T_{h} in terms of the
Lagrange multiplier . We then reduce our problem to the
resolution of a linear system with explicit coefficients in terms
of the geometry of the triangulation with only the Lagrange multiplier
on each edge and the discrete pressure
p_{h} on each triangle as unknowns.
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