7.1.3.1 Data distribution

Let us denote by $\bullet$ the nodes of the velocity and by $\square$ those nodes belonging to the pressure. We will use the following elements in the discretization :

(a)		non-conform [P$_1$, P$_{0^-}$] element
(b)		conform (in $u,p$) [P$_1$+bubble, P$_1$] MINI-element
(c)		conform (in $u$) [Q$_1^h$, Q$_0^H$] element, pressure components are located in the center of a macro element

The non-overlapping element distribution from section 4.3.1 seems to be best suited for the sake of parallelization. In case of element type (c) one should distribute the macro elements.
All matrices in (7.4) are calculated element-wise so that the distributed storing of them is obvious, i.e., ${\ensuremath{\color{green}{\sf M}}} = \sum\limits_{s=1}^p A_s^T {\ensuremath{\color{green}{\sf M}}}_s A_s$ and similar ${\ensuremath{\color{green}{\sf D}}}$, ${\ensuremath{\color{green}{\sf C}}}$, ${\ensuremath{\color{green}{\sf B}}}$ (and even the vector $\underline{{\ensuremath{\color{green}{\sf f}}}}$). These matrices are calculated and stored in the subdomains so that normally there will be no communication in the matrix generation. The only exception may occur in the calculation of the convection matrix  ${\ensuremath{\color{green}{\sf C}}}_s$ if upwind-schemes of order higher than 1 are used. $\left(\sum\right)$] $\left(\sum\right)$ Here, the necessary knowledge of the velocity components of others than next neighbor elements requires a preparing communication step.