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\begin{center}
ON A CONVERGENCE OF AN APPROXIMATE SOLUTION 
FOR A SOLID-FLUID MODEL

\vspace*{0.2cm}
{\it G. Chichua}

\vspace*{0.2cm}
{\it 
I.Vekua Institute of Applied Mathematics}
\end{center}

\vspace*{0.15cm}
\par
This paper deals with the study of the approximation scheme for a boundary-contacts
problem with unilateral contact conditions for the body containing parts of 
elastic solid and viscous fluid (see [1], [2]).
\par
Let vector $h=(h_1,h_2,h_3)\in {\Bbb R}^3$. We define the mesh $R_h$ (see [3])by 
\begin{equation}
R_h=\{M\vert M\in {\Bbb R}^3,\;\;
M=(m_1h_1,m_2h_2,m_3h_3),\;\;m_i\in Z\}.
\end{equation}
\par
To any node $M$ of the mesh $R_h$ we correspond a "bar" with the center $M$:
$$
\omega^0_h(M)={\mathop\prod\limits_{i=1}^3}](m_i-\frac{1}{2})h_i,
(m_i+\frac{1}{2})h_i[
$$
and a "christ" with the center $M$:
$$
\omega^1_h(M)={\mathop\cup\limits_{i=1}^3}
\omega^0_h(M\pm \frac{h}{2}e_i),
$$
where $e_i,\;\;i=1,2,3$ are unit basis vectors in ${\Bbb R}^3$
\par
Let $A=\{M\in R_h\vert dist(M,\sum)<\frac{1}{2}\sqrt{h_1^2+h_2^2+h_3^2}\}$. 
If $M\in A$, then we introduce $(\omega^0_h(M))'=\omega^0_h(M)\cap\Omega_s$ and 
$(\omega^0_h(M))''=\omega^0_h(M)\cup\Omega_f$.
\par 
Now we define $\Omega_h=\{M\vert \omega^1_h(M)\subset\Omega\}$ and $v_h$ which is the space streched on $\theta_h^M,\;
(\theta_h^M)',\;(\theta_h^M)'',\;\;M\in\Omega_h$,
where the last three symbols are characteristic functions respectively for 
$\omega^0_h(M)$, when $dist(M,\sum)\ge\frac{1}{2}\sqrt{h_1^2+h_2^2+h_3^2}$ 
and $(\omega^0_h(M))'$
and $(\omega^0_h(M))''$.
\par
We assume that
\begin{equation}
K_h=\left\{v_h\vert v_h\in V_h:(v_h,(\theta_h^M)')-(v_h,(\theta_h^M)'')\ge 0\;\;
{\rm a.e.\;\;on}\sum\right\}
\end{equation}
\par
The following scheme of approximation has been built:
find $ u^2,u^3,...u^n $, such, that 
\begin{equation}
(\rho\gamma^i,v-d^i)+a(u^{i+1},v-d^i)+b(d^i,v-d^i)\ge (f^i,v-d^i),\;(i\ge 1)
\end{equation}
\begin{equation}
\begin{array}{c}
\forall v\in K_h,\;\; d^i\in K_h,
\\
u ^0=u^1=0,
\end{array}
\end{equation}
where we use the following notations (we do not write $ h $ index):
\begin{equation}
\delta^i=\frac{u^{i+1}-u^i}{k},
\end{equation}
\begin{equation}
d^i=\frac{u^{i+1}-u^{i-1}}{2k}=\frac{\delta^{i}+\delta^{i-1}}{2},
\end{equation}
\begin{equation}
\gamma^i=\frac{\delta^{i}-\delta^{i-1}}{k}=
\frac{u^{i+1}-2u^i+u^{i-1}}{k^2}=
2\left(\frac{d^i-\delta^{i-1}}{k}\right),
\end{equation}
\begin{equation}
g^i=\frac{f^{i+1}-f^i}{k}.
\end{equation}
\par
Taking into account these notations, we can rewrite the inequality (3) in
the following way
$$
(\rho d^i,v-d^i)-(\rho\delta^{i-1},v-d^i)+
k^2a(d^i,v-d^i)+\frac{k}{2}a(u^i-k\delta^{i-1},v-d^i)+
$$
\begin{equation}
+\frac{k}{2}b(d^i,v-d^i)-\frac{k}{2}(f^i,v-d^i)\ge 0.
\end{equation}
\par
The problem (9), (4) can be reduced to the problem of minimization of the
following functional:
\begin{equation}
J(d^i)=\frac{1}{2}\alpha\vert d^i\vert^2+\frac{1}{2}k^2a(d^i,d^i)+
\frac{k}{4}b(d^i,d^i)+(\tilde{f_i},d^i),\;\;d_i\in K_h,
\end{equation}
\par
\begin{equation}
where\;(\tilde{f_i},d^i)=-(\rho\delta^{i-1},d^i)+
\frac{k}{2}a(u^i-k\delta^{i-1},d^i)-(f^i,d^i),
\end{equation}
\begin{equation}
u^{i+1}=u^i+kd^i
\end{equation}
Let $ P_k:V_h\to K_h $  be a mapping defined as follows:
$$
P_k(v_h)=
\cases{v_h\;\;if\;\;(v_{h},(\theta_h^M)'')-
(v_{h},(\theta_h^M)')\ge 0,\;\;\forall M\in A,
\cr v_h\;\;if\;\;(v_{h},(\theta_h^M)'')-(v_{h},(\theta_h^M)')
< 0,\;\;\forall M\in A,}\;\;v_h\in V_h,
$$
where $ (v^1_{h},\theta_h^M)=
(v_{h},\theta_h^M) $ if $M\in\Omega_h\backslash A$, and
$ (v^1_h,(\theta_h^M)')=
(v_h^1,(\theta_h^M)'')=(v_h,\theta_h^M)'$ if $ M\in A $.
\par
{\bf Theorem.} {\it Let $$ d^0\in K_h ,\; d^{n+1}=P_k(d^n-\rho_nJ'(d^n)),
\eqno{(13)}$$
where  $ \rho_n $ are numbers.}
\par
We can choose the numbers $ \rho_0 $ and $ \rho_1 $ such, that if
$ 0<\rho_0\le\rho_n\le\rho_1 $, than the process of iteration (13) 
converges to the solution of the minimization problem (10).

\newpage
\footnotesize
\begin{center}
{\bf R e f e r e n c e s}

\vspace*{0.2cm}
\end{center}
\par
[1] G. Chichua. On a Boundary-contact Problem for a Solid-Fluid Model. Reports
     of Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics, 1995, v. 10, $N^{_{\underline 0}}$1,
p. 18-20.

[2] G. Chichua. Mixture of Elastic Solid and Viscous Fluid. 
Tbilisi International Centre of Mathematics and Informatics, 1997, v.1.

[3] Sanches-Palencia E. Non-Homogenous Media and Vibration Theory. Springer-Verlag, 1980.
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