5.7.2.3 Smoothing

The smoothing operators $S_{_{\mathrm{pre}}}$ and $S_{_{\mathrm{post}}}$ like $\omega$-Jacobi, Gauß-Seidel and ILU (5.3 - 5.5) require as input an accumulated initial solution  $\underline{{\ensuremath{\color{red}\mathfrak{u}}}}^0$, a distributed stored right hand side  $\underline{{\ensuremath{\color{red}\mathfrak{u}}}}^0$ and a distributed matrix  ${\ensuremath{\color{green}{\sf K}}}$. The iterated solution  $\underline{{\ensuremath{\color{red}\mathfrak{u}}}}$ is accumulated in any case. Each iteration of a smoother requires at least one accumulation.

$\Longrightarrow$

Vector  $\underline{{\ensuremath{\color{red}\mathfrak{u}}}}_{q}$ is accumulated, $\widehat{\underline{{\ensuremath{\color{red}\mathfrak{u}}}}} \makebox[0pt]{}_{q}$ and $\widetilde{\underline{{\ensuremath{\color{red}\mathfrak{u}}}}} \makebox[0pt]{}_{q}$ are accumulated in any case.

$\Longrightarrow$

Vector  $\underline{{\ensuremath{\color{red}\mathfrak{w}}}}_{q}^0$ is also accumulated.

Communication in smoothing !