7.1.3.2 Parallel fix point iteration

All submatrices in (7.4) are distributed ones (see section 4.3.1). So, defining all state variables $\underline{{\ensuremath{\color{red}\mathfrak{u}}}}$, $\underline{{\ensuremath{\color{red}\mathfrak{p}}}}$ as accumulated vectors and all integral variables / functionals $\underline{{\ensuremath{\color{green}{\sf f}}}}$ ( $\underline{{\ensuremath{\color{green}{\sf r}}}}$, $\color{green}{\sf s}$) as distributed vectors is again obvious.

Here,

$${\ensuremath{\color{green}{\sf\widehat{f} \makebox[0pt]{}}}} := \sigma {\ensuremath{\color{green}{\sf f}}}(t_{K+1}) + (1-\sigma) ... ...u}}}}^{K}) \right) \right] \underline{{\ensuremath{\color{red}\mathfrak{u}}}}^K$$

$${\ensuremath{\color{green}{\sf A}}}(\underline{{\ensuremath{\color{red}\mathfrak{u^n}}}}) := \tfrac{1}{\tau} {\ensuremath{\color{green}{\sf M}}} + \sigma \lef... ...sf C}}}(\underline{{\ensuremath{\color{red}\mathfrak{u}}}}^n)}\right) \enspace.$$

• Parts of the algorithm which require communication are marked by $\fbox{\qquad }$ or the accumulation sum $\sum\limits_{s=1}^p$.
• In the inner product, a change of the type for the accumulated vectors $\underline{{\ensuremath{\color{red}\mathfrak{u}}}}^{n}_{\delta,s}$ and $\underline{{\ensuremath{\color{red}\mathfrak{p}}}}^{n}_{\delta,s}$ is necessary. This can be done without communication (4.6).
  If element types (a) or (c) with distribution of macro elements are used then $R_{{\ensuremath{\color{red}\mathfrak{p}}}} = I_{N_p}$.