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\begin{center}
THE INVESTIGATION OF THE STABILITY OF THE MULTISTEP METHOD 
FOR THE OPERATOR VALUED ABSTRACT PARABOLIC EQUATION WITH
THE USE OF ASSOCIATIVE POLYNOMIALS

\vspace*{0.2cm}
{\it J. Rogava, R. Galdava}

\vspace*{0.1cm}
{\it Georgian Technical University }
\end{center}

\vspace*{0.1cm}\par
Let us consider the Cauchy problem in the  Hilbert space $H$:
\begin{equation}
  u'(t)+A(t)u(t)=f(t), \ \ t > 0,  \ \  u(0)=\vf,
\end{equation}
where $A$ is a self-adjoint positive definite operator with domain of 
definition $D(A)$ that is independent of $t$; $f(t)$ is an abstract function
with values from the space $H$; $\vf$ is an initial vector; $u(t)$ is
the sought for function.

Consider the multistep method for problem $(1)$, which is 
described by the following scheme:
\begin{equation}
 \sum\limits_{i=0}^q \alpha_i u_{k-i} + \tau A(t_k) u_k = \tau f(t_k),
\end{equation}
where
      $$ k=q,q+1, \cdots , (q \ge 2), \ \ t_k =k \cdot \tau \ \ (\tau > 0), \ 
            \ \alpha_0 >0, $$
      $$ \sum\limits_{l=1}^q \alpha_i = 0. $$ 


{\bf Theorem. } {\it Assume the following :

$a)$ \ $A(t)$ is a self-adjoint positive definite operator for any $t$ on 
     $[0,+\infty [ $ ;
     
$b)$ for any $t'$, $t''$ and $s$ from  $[0,+\infty[$, the following
     inequality holds 
 
$$\|[A(t')-A(t'')]A^{-1}(s)\| \le c_0 |t'-t''|, \ \ c_o=const>0; $$

$c)$ for any $x$ on $[0,1]$ all the roots of the characteristic equation
\begin{equation}
    \alpha_0 \lb^q + \alpha_1 x^{q-1} \lb^{q-1} + \cdots + \alpha_{q-1} x \lb 
    + \alpha_q =0
\end{equation}
possibly, except one root, belong to the some circle, which is settled
within the unit circle and the exceptional root belongs to the unit circle,
or all the roots of the equation $(3)$, possibly, 
with the exception of two of them, belong
to the same circle, which is settled within the unit circle, and the residual
roots $\lb_1 (x)$ and $\lb_2 (x)$ belong to the unit circle and satisfy the
following conditions: $1) \lb_1 (x)=\overline{\lb_2 (x)}$ or both are real;
$2)$ there is a number $0< \dl <1 $ that is independent of $x$,
the distances from the point $(\lb_1 +\lb_2;\lb_1 \lb_2 )$ to the points
$(2;1)$ and $(-2;1)$ is more or equal to $\dl$.
 Then the true estimate for the scheme $(2)$ is the following one:
\begin{equation}
      \| u_{k+q-1} \| \le c [e^{ct_{k-1}}
     (\| u_0 \|+ \| u_1 \|+ \cdots + \|u_{q-1} \| )+
     \tau \sum\limits_{i=1}^k e^{ct_{k-i}} \| f(t_{i+q-1}) \| ] 
\end{equation}   
where  $k=q,q+1, \cdots , (q \ge 2) ,\ \ \ c=const>0. $
}
\par{\bf The scheme of the proof.}

From $(2)$ it follows:
\begin{equation}
        u_k = \bt_1 L_k u_{k-1} + \bt_2 L_k u_{k-2} + \cdots +
              \bt_q L_k u_{k-q} + \tau \bt_0 L_k f( t_k ), 
\end{equation}
where
$$ \bt_0 = \frac{1}{\alpha_0 }, \ \  \bt_i = - \frac{\alpha_i}{\alpha_0} ,
           \ \  i=1,2, \cdots ,q,$$
$$ L_k = (I+\tau \bt_0 A(t_k)) ^{-1}, \ \ k=q,q+1, \cdots  $$

By the method of induction the following equation is obtained from  $(5)$:
$$
 u_{k+q-1}=(\bt_1 \skl{q}{U}_{k-1} L_q + \bt_2 \skl{q+1}{U}_{k-2} L_{q+1} +
           \cdots + \bt_q \skl{2q-1}{U}_{k-q} L_{2q-1})u_{q-1}+
$$
$$
         + (\bt_2 \skl{q}{U}_{k-1} L_q + \bt_3 \skl{q+1}{U}_{k-2} L_{q+1} +
           \cdots + \bt_q \skl{2q-2}{U}_{k-q+1} L_{2q-2})u_{q-2}+
$$
$$
         + (\bt_3 \skl{q}{U}_{k-1} L_q + \bt_4 \skl{q+1}{U}_{k-2} L_{q+1} +
           \cdots + \bt_q \skl{2q-3}{U}_{k-q+2} L_{2q-3})u_{q-3}+
$$
\begin{equation}
   + \cdots + \bt_q \skl{q}{U}_{k-1} L_q u_0 + 
    \tau \bt_0 \sum\limits_{i=1}^k \skl{i+q-1}{U}_{k-i} L_{i+q-1} f(t_{i+q-1}),
\end{equation}
where $\skl{i}{U}_k$ operators are determined by the following recurrent dependence:
\begin{equation}
 \skl{i}{U}_k = \bt_1 L_{k+i} \skl{i}{U}_{k-1} + \bt_2  L_{k+i} \skl{i}{U}_{k-2}
   + \cdots + \bt_q  L_{k+i} \skl{i}{U}_{k-q},      
\end{equation}
    $$ \skl{i}{U}_0=I,\ \ \skl{i}{U}_{-1}= \skl{i}{U}_{-2}=
                  \cdots =\skl{i}{U}_{1-q} =0. $$
 
If the operators $\skl{i}{U}_k$  are uniformly bounded, then according to  
equation $(6)$ a priori estimate $(4)$ holds,thereby the scheme 
$(2)$ is stable, and on the contrary, if the scheme $(2)$ is stable, then the 
operators $\skl{i}{U}_k$ are uniformly bounded (in this case we assume that 
$t$ is varying in a finite interval).

Let's see if the $\skl{i}{U}_k$ operators are uniformly bounded when the 
conditions of the theorem hold. To prove this fact the operator-valued 
polynomials are necessary, which are determined by the following 
recurrent dependence:
$$
        U_k (\bt_1 L, \bt_2 L, \cdots , \bt_q L)=
        \bt_1 L U_{k-1} (\bt_1 L, \bt_2 L, \cdots , \bt_q L)+
$$
\begin{equation}
      + \bt_2 L U_{k-2} (\bt_1 L, \bt_2 L, \cdots , \bt_q L)+ \cdots +
        \bt_q L U_{k-q} (\bt_1 L, \bt_2 L, \cdots , \bt_q L),      
\end{equation}
$$ U_0=I,\ \ \ \ U_{-1}=U_{-2}= \cdots = U_{1-q} =0, $$
$$    L=(I+\tau \bt A(t_j))^{-1}.  $$
These polynomials are called polynomials associated to the scheme $(2)$.

The scheme $(2)$ for an operator $A$ which does not depend on $t$ 
with the use of associated polynomials was investigated in $[6]$.

The operators $\skl{i}{U}_k$ are
related to the operator-valued polynomials $ U_k (\bt_1 L,$ $ \bt_2 L,$
$ \cdots , \bt_q L)$ 
by the following equation:
\begin{equation}
  \skl{q+j-1}{U}_k = U_k - \tau \bt_0 \sum\limits_{i=1}^k
  \skl{i+q+j-1}{U}_{k-i}  L_{i+q+j-1} [ A(t_{i+j}) - A(t_j) ] U_i .
\end{equation}

The following estimates hold:
\begin{equation}
      \|  U_k (\bt_1 L, \bt_2 L, \cdots , \bt_q L) \| \le c;            
\end{equation}
\begin{equation}
      \|  U_k (\bt_1 S^{q-1}, \bt_2 S^{q-2}, \cdots , \bt_q I) \| \le c;            
\end{equation}
\begin{equation}
      \| \tau A(t_j) S^k \| \le \frac{q}{\bt_0 k},            
\end{equation}
where $ S=L^{\frac{1}{q}} $ , \ \ \ $c=const>0.$
\par Taking into account
$$
U_k\left(\beta_1L,\beta_2L,...,\beta_qL\right)=L^{\frac{k}{q}}
U_k\left(\beta_1S^{q-1},\beta_2S^{q-2},...,\beta_qI\right),
$$
according to inequalities $(11),\;(12)$ and the  
conditions of the theorem the following is true:
$$
  \|( A(t_{k+j}) - A(t_j)) U_k (\bt_1 L, \bt_2 L, \cdots , \bt_q L) \| \le
$$           
$$
  \le  \|( A(t_{k+j}) - A(t_j)) A^{-1} (t_j) \| \| A(t_j) L^{\frac{k}{q}} \|
       \| U_k (\bt_1 S^{q-1}, \bt_2 S^{q-2}, \cdots , \bt_q I) \| \le            
$$
\begin{equation}
  \le c_0 \frac{t_{k+j} - t_j}{\tau } \cdot \frac{q}{\bt_0 k} \cdot c= c,\;\;
c={\rm const}>0.
\end{equation}
With the help of estimates  $(10)$ and $(13)$ from $(9)$, we obtain
\begin{equation}
   \| \skl{q+j-1}{U}_k \| \le c + c \tau \sum\limits_{i=1}^k
     \| \skl{i+q+j-1}{U}_{k-i} \| .
\end{equation}
Applying the discrete analog of the Gronwall lemma the following 
estimate can be obtained:  
\begin{equation}
   \| \skl{i+q+j-1}{U}_{k-i}  \| \le c e^{ct_{k-i}}.
\end{equation}

Considering this estimate a priori estimate  $(4)$ follows from 
equation $(6)$.

The proof is complete. %$\square$

To the investigation of multistep method for an abstract parabolic 
equation are dedicated the papers $[1]-[5]$.


\vspace*{0.2cm}
\footnotesize
\begin{center}
{\bf R e f e r e n c e s}

\vspace*{0.1cm}
\end{center}
\par
[1] Crouzeix M., Une Methode Multipas Implicite-explisite pour l'approximation 
des Equations d'evolution Paraboliques. - Numerische Mathematik, 35, 1980,
pp.257-276.

[2] Crouzeix M., Raviart P.A., Approximation des Equations d'evolution Lineaires par
des Methodes Pasmultiples.- C.R.Acad. Sci. Paris, 283, A, 1976, 367-370. 

[3] Le Roux M.N., Semi-Discretisation en Temps pour les Equations d'evolution  
Para\-bo\-li\-ques Lorsque l'operateur Depend du Temps. - R.A.I.R.O. Analyse 
Numerique/Numerical Analisis, 13, 1979, 119-137. 

[4] Zlamal M., Finite Element Multistep Methods for Parabolic Equations. 
I.S.N.M. 28, Birkh${\mathop{\rm a}\limits^{..}}$tuser Verlag. Basel and Stuttgart, 1975.

[5] Zlamal M., Finite Element Multistep Discretisations of Parabolic Boundary Value
Problems. - Math. Comp., 29, 350-359.

[6] Rogava J.L., Semi-discrete Schems for Operator Differential Equations. 
Tbilisi: Technical University press, 1995, (Russian).


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