Abstract:
This paper is an investigation of a group of iterative methods with complete boundary-condition splitting for solving the first boundary value problem for a system of Stokes type with a small parameter $\varepsilon>0$:
\begin{gather*}
-\varepsilon ^2\Delta{\mathbf u}+{\mathbf u}+\operatorname{grad}p={\mathbf f},
\qquad \operatorname{div}{\mathbf u}=0\quad \text {in </nomathmode><mathmode>$\Omega $},
{\mathbf u}|_\Gamma ={\mathbf g}, \qquad \int _\Gamma ({\mathbf g},{\mathbf n}) ds=0,
\end{gather*} </mathmode><nomathmode>
where $\mathbf{u}=(u^1(x),\dots,u^n(x))$ is the velocity vector, $p = p(x)$ is the pressure, $\mathbf{f}=(f^1(x),\dots,f^n(x))$ is the field of external forces, and
$\mathbf{g}=(g^1(x),\dots,g^n(x))$ is a given value of the velocity vector on the boundary $\Gamma$ of a domain $\Omega$ in the $n$-dimensional Euclidean space
$\mathbb{R}^n$.
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