UNIVERSITY OF COLORADO AT DENVER

PLACE: Mathematics Conference Room 656 UCD Building, 1250 14th St., Denver

TIME: 12:45 pm (Refreshments served at 12:30 pm)

DATE: May 4, 1999

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\centerline{\bf A minimal stabilisation procedure for mixed finite element methods.}
\centerline{\it by F.Brezzi, M.Fortin}
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{\bf Abstract} Let us consider the classical variational problem in {\it
mixed form}: find $u\in V$ and $p\in Q$ such that
$$ a\,(u,v)+b\,(v,p)-b\,(u,q)=-~~\forall \, v\in V,~\forall\, q\in Q,$$
where $a\,(\,,\,)$ and $b\,(\,,\,)$ are bilinear continuous forms on $V\times V$ and $V\times Q$ respectively and $f,g$ are given in $V^{\prime}$ and $Q^{\prime}$
respectively.\par It is well known that, when discretising problems of this type,
the finite element spaces $V_h\subset V$ and  $Q_h\subset Q$ cannot be chosen arbitrarily but have to satisfy suitable {\it compatibility conditions}. If, 
for some reason, one has a definite need for using a specific pair of subspaces
which are incompatible (for instance, when you have to use the same variables
in other equations, within a bigger system) powerful techniques are available
(usually called {\it \`a la Hughes-Franca}) that allow to stabilize, virtually,
every given choice of $V_h$ and $Q_h$. In certain cases, however, this amounts
in some sense to overkill the problem, and makes the choice of the stabilizing parameters a delicate one.\par The aim of the present technique (that we called {\it minimal stabilisation}) is to recognise that, in a number of cases, a given choice
that is not stable has nonetheless a certain amount of stability. In other words, only a subspace of $V_h\times Q_h$ will be out of control, while the
rest would have a tendency to behave properly. The idea is then to identify,
whenever possible, the stable part, and then apply a stabilization that acts only on the remaining part. \par
A general abstract theory for these type of techniques will be presented, and
classes of applications, mostly on Stokes problem, will be pointed out. These
will include the known tricks for stabilizing $Q1-P0$ and $P1-P0$, as well
as the so called Habashi-stabilization of the $P1-P1$ {\it equal order} approximation. 

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