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ON A COUPLING OF DIFFERENTIAL EQUATIONS VIA INTERFACE BOUNDARY CONDITIONS

\vspace*{0.3cm}
{\it R. Bochorishvili, D. Jaiani}

\vspace*{0.3cm}
{\it 
I.Vekua Institute of Applied Mathematics}
\end{center}

\vspace*{0.3cm}
\par
It is well known that different mathematical models can be used for the description
of the same phenomena. Of course each of these models has own range of validity
and the relationships between dependent variables of different models are known.
As usual, the simple models are easier to solve but they have restricted areas of
validity. More comlicated models can be successfuly applied to broader areas but
they require more efforts to be solved. From computational point of view, naturally,
it is expedient to make adaptation by mathematical models with the purpose to use
simpler models where it is possible. So, if we study some problem in $\Omega$
then there may exist different sub domains of $\Omega$ where different mathematical
models are most suitable. In this case one can make adaptation by mathematical models,
i.e. use different models in different sub domains and set relationships between
dependent variables of different mathematical models as interface boundary conditions.
With account of the above the problem can be formally formulated as follows:
\begin{equation}
\alpha_i\frac{\partial^{k_i}u_i}{\partial t^{k_i}}+A_iu_i=f_i\;\;\;{\rm in}\;\;(0,T)\times \Omega_i,
\end{equation}
\begin{equation}
\frac{\partial^{\ell}u_0}{\partial t^{\ell}}=u_{i\ell}\;\;\;{\rm in}\;\;
\Omega_i\;\;{\rm at}\;\;t=0,\;\;\ell=0,1,...,k-1;
\end{equation}
\begin{equation}
L_iu_i=\varphi_i\;\;\;{\rm on}\;\;(0,T)\times \gamma_i;
\end{equation}
\begin{equation}
L_{ij}u_i=L_{ji}u_j+\varphi_i\;\;\;{\rm on}\;\;(0,T)\times \gamma_{ij},
\end{equation}
where $i=1,2,...,m$, $j=1,2,...,m$, $k_i=const,$ $\alpha_i=const$,
\begin{equation}
\begin{array}{c}
\overline{\Omega}={\mathop\cup\limits^m_{i=1}}
\overline{\Omega}_i,\;\;\gamma=\partial\overline{\Omega},\;\;
\gamma_i=\gamma\cap{\overline\Omega_i},\\
\Omega_i\cap\Omega_j=\O\;\;{\rm if}\;\;i\ne j,\;\;
\gamma_{ij}={\overline\Omega_i}\cap{\overline\Omega_j}\;\;
{\rm when}\;\;i\ne j,
\end{array}
\end{equation}
$f_i$, $u_{i\ell}$, $\varphi_i$, $\varphi_{ij}$ are given functions, $A_i$,
$L_i$, $L_{ij}$, $L_{ji}$ are given operators. Solution of the problem $u$
is defined as $u_i$ on $(0,t)\times\Omega_i$. In the formulaes given above (1)
correspond to different equations, (2) are initial conditions, (3) are boundary conditions,
(4) are interface boundary conditions.
\par
The following proposition shows that (1)-(4) is not a correct problem.
\par
{\bf Proposition 1.} if
\begin{equation}
L_{ij}u_i={\tilde\varphi}_{ij}\;\;\;{\rm on}\;\;j\in I_i,\;\;
I_i=\{j,\;\gamma_{ij}\ne \O\}
\end{equation}
and the problem (1)-(3), (6) is correct for any $i$ fixed then the coupled problem
(1)-(5) has an infinite number of solutions.
\par Of cource we suppose that ${\tilde\varphi}_{ij}$ are taken from suitable function spaces. Note
that updating ${\tilde\varphi}_{ij}$ on ${\gamma}_{1j,\;j\in I_1}$, yields correct problems for $u_1$ and boundary conditions
for j-th equations that can be calculated from (4). Clearly, on $\gamma_{2j}$,
$j\in I_2$, we already have boundary conditions on $\gamma_{2}$ and $\gamma_{12}$,
if $\gamma_{2}\ne 0$, $\gamma_{12}\ne \O$. For other boundaries of $\Omega_2$
arbitrary functions ${\tilde\varphi}_{2j}$ can be updated. Repeating the same
procedure for other sub domains we obtain correct problems for all 
$ i=1,2,...,m $. Since some $ \tilde\varphi_{ij} $ functions were updated
arbitrarily, validity of the proposition is evident.
\par
Thus with account of the above proposition it is necesarry to modify the 
coupled problem in such a way that it be correct. We suppose that for this
sake the following approaches can be used:
\par
(i) To overlap a part of domains $ \Omega_i $ and $\Omega_j $,
$ \Omega_{ij}=\Omega_i\cap\Omega_j,\;i\ne j $, meas\\ $ \Omega_{ij}\ne 0;$
to consider each of problems (1)-(3) in $ \Omega_i\cup\Omega_{ij} $ and
$ \Omega_j\cup\Omega_{ij} $ correspondingly and to set additional requirements
$$
\mathop{\int}\limits_{\Omega ij}\vert L_{ij}u_i-L_{ji}u_j-\varphi_{ij}\vert
d\Omega\to min.
$$
\par
Such an approach was considered in [1] for coupling viscous and inviscid 
models of incompressible fluid.
\par
(ii) To add or derive somehow additional conditions on interface boundaries 
$ \gamma_{ij} $ as for example it was made in [2] for coupling various 
models for transonic flow computation.
\par
(iii) To overlap domains as in (i) and to set interface boundary conditions 
(4) on $ \partial\Omega_{ij} $.
\par
(iv) To select interface boundary in special way within 
$ (0,T)\times\Omega_{ij} $.
\par
Below the last two approaches are demonstrated for the following simple problem:
coupling of second order hyperbolic and elliptic equations in one space 
dimension. Namely, coupled according to (1)-(4) problem is the following:
\begin{equation}
\frac{\partial^2u_1}{\partial t^2}=
a^2\frac{\partial^2u_1}{\partial x^2}\;\;in\;\;
(0,T)\times\Omega_{1},\;\;\Omega_1=(0,1),
\end{equation}
\begin{equation}
u_1(0,x)=u_{10}(x),\;u_{1t}(0,x)=u_{11}(x),\;x\in\Omega_1,
\end{equation}
\begin{equation}
u_1(t,0)=0;
\end{equation}
\begin{equation}
\frac{\partial^2u_2(t,x)}{\partial x^2}=0\;\;in\;\;
(0,T)\times\Omega_{2},\;\;\Omega_2=(1,2),
\end{equation}
\begin{equation}
u_2(t,2)=0,
\end{equation}
\begin{equation}
u_2(t,1)=\frac{\partial u_1(t,1)}{\partial x}.
\end{equation}
\par
According to approach (iii) let us introduce domain of overlapping as 
$ \Omega_{12}=$ $=(1,1+b) $ and set the corresponding problem:
\begin{equation}
\frac{\partial^2u_1}{\partial t^2}=
a^2\frac{\partial^2u_1}{\partial x^2}\;\;on\;\;
(0,T)\times(\Omega_{1}\cup\Omega_{12}),
\end{equation}
\begin{equation}
u_1(0,x)={\tilde u}_{10}(x),\;u_{it}(0,x)={\tilde u}_{11}(x)\;
\;in\;\Omega_1\cup\Omega_{12},
\end{equation}
\begin{equation}
u_2(t,1)=\frac{\partial u_1(t,1)}{\partial x},\;\;
u_2(t,1+b)=\frac{\partial u_1(t,1+b)}{\partial x}.
\end{equation}
\par
Note that (10)-(11), (13)-(15) correspond to coupled problem according to the
approach (iii) given above,
\begin{equation}
{\tilde u}_{10}(x)=\cases{u_{10}(x),\;x\in\Omega_1,\cr
u_{2}(0,x),\;x\in\Omega_2,}
\end{equation}
\begin{equation}
{\tilde u}_{11}(x)=\cases{u_{11}(x),\;x\in\Omega_1,\cr
u_{2t}(0,x),\;x\in\Omega_2,}
\end{equation}
where $ u_2(0,x),u_{2t}(0,x) $ are solutions of (9) with boundary
conditions (10) and 
\begin{equation}
u_2(0,1)=u_{10}(1)\;\;or\;\;
u_{2t}(0,1)=u_{11}(1)
\end{equation}
correspondingly.
\par
The following proposition hold true:
\par{\bf Proposition 2.} In case of sufficiently smooth initial data satisfying 
consistency conditions classical solution of coupled problem 
(10)-(11), (13)-(18) exists and is unique.
\par
In order to demonstrate approach (iv) let us consider coupled problem of
the type (7)-(12) but with interface boundary curve $ \gamma_{12} $,
where $ \gamma_{12} $ lies in $ (0,T)\times{\overline\Omega}_{12} $.
This means that on the left of $ \gamma_{12} $ one has problem similar
with (7)-(9), on the right of $ \gamma_{12} $ one has problem similar
with (10),(11)  and on $ \gamma_{12} $ one has interface boundary condition
similar with (12). For such a problem the following proposition holds true:

{\bf Proposition 3.} In $ (O,T)\times\Omega_{12} $ there exists an infinite number of
interface boundary curves $ \gamma_{12} $ for which solution of the 
corresponding coupled problem exists and is unique.

\vspace*{0.5cm}
\footnotesize
\begin{center}
{\bf R e f e r e n c e s}

\vspace*{0.3cm}
\end{center}
\par
[1] Dinh Q.V., Glowinski R., Periaux J., Terrasson G.,
   On the Coupling of Viscous and Inviscid Models for Incompressible
   Fluid Flows Via Domain Decomposition. SIAM, Philadelphia, pp. 350-369.
\par
[2] Berger H., Warnecke G., Wendland W.L., Analysis of a FEM/BEM Coupling Method 
   for Transonic Flow Computations. Preprint 93-9, Mathematisches Institut A 
   der Universitat stuttgart, 1993, pp. 1-44.

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