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El\.zbieta Motyl

Cracow University of Technology

Institute of Mathematics
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\subsection*{ STABILITY OF THE HOLLY METHOD FOR   \\
      THE STATIONARY NAVIER-STOKES EQUATIONS }
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We are concerned with the following stationary Navier-Stokes problem in a bounded
domain $\Omega \subset {| \! \!  R}^n $ ($ n \in \{ 2, 3 \} $) with boundary
$\partial \Omega $ satisfying the Lipschitz condition
\begin{displaymath}
(N-S) \left\{
      \begin{array}{cl}
     & \sum_{i=1}^{n} v_i \frac{\partial v}{\partial x_i} = \nu \Delta v + f -
        \bigtriangledown p , \\
     & divv = 0 , \\
     & v_{\partial \Omega }=0 ,
     \end{array}
     \right.
\end{displaymath}
where $\nu \in ]0, \infty [ $ (viscosity), $f: \Omega \to {|\! \! R}^n $ (external forces)
are given, while velocity $v: \Omega \to {|\! \! R}^n $ and pressure $p: \Omega \to
 | \! \! R$ are looked for.
We consider weak solutions of this problem in the sense of J.Leray.

From the numerical point of view  application of the Faedo-Galerkin method requires
the internal approximation of the space $V$ of solenoidal vector fields of the
Sobolev space $H^1_0 $. The incompressibility condition makes it difficult to construct
such an approximation. In general, this problem has not been solved yet succesfully,
while the space $H^1_0 $ is well approximated, e.g. by splines in the finite element
method.

Prof. K.Holly introduced a new numerical method for the (N-S) problem, which does
not require the internal approximation of $V$.
This method is based on the approximation of the exact solutions by the sequences
$(v_s)_{s \in |\! \! N} $ and $(v_{s,N})_{N \in |\! \! N} $  of solutions of appropriate
equations in $H^1_0 $ and  in finite dimensional subspaces of $H^1_0 $, respectively.

We prove the stability of this method in the sense that for the pair
(velocity, external forces ) from some generic set $\cal G$, the sets of solutions
of successiv approximate problems for sufficiently large $s, N \in |\! \! N $ and the
set of exact solutions are finite, equinumerous and convergent in the Hausdorff
metric. This results guarantee that for $(\nu , f ) \in \cal G $ each solution
of the (N-S) problem is attainable by the Holly method. The fact that this sets
are equinumerous reduces the problem of uniqueness of the (N-S) equations
to the finite dimensional case for sufficiently large $s, N \in | \! \! N$.



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