7.2.4 Parallelization using split boxes

Let us represent the volume elements by their centers of gravity (with the unknowns stored therein) and connect them by lines. This procedure results in a kind of FEM mesh which we can distribute in the same way as the distributed elements from section 4.3.1. Therefore, all conclusions on matrix and vector types of that section can be applied.
We choose the functionals  ${\ensuremath{\color{green}{\sf g}}}$ as distributed stored and the unknowns  ${\ensuremath{\color{red}\mathfrak{u}}}$ as accumulated vector. Note that the values $\boldsymbol{\sigma}_{\boldsymbol{pl}}$ needed in the calculation of  ${\ensuremath{\color{green}{\sf g}}}$ are already accumulated and must not counted twice. On the other hand, the contribution of flux  ${\ensuremath{\color{green}{\sf g}}}_{jl}$ is an integral value over the edge $S_{jl}$. So using in (7.18) only that part of the edge belonging to $\Omega_j$ (denoted by ${\ensuremath{\color{green}{\sf s}}}_{jl}$) solves this contradiction and we can calculate the contributions  ${\ensuremath{\color{green}{\sf g}}}_{jl}$ locally.

Figure 7.4: Split boxes

Remarks :

1. Besides the initial calculation of the volumes, algorithm 7.9 requires only one communication per time step.
2. All data needed in the calculation of $C_1$ and $C_2$ are locally accessible so that no additional communication is necessary.
3. Analogously to algorithm 7.9, arithmetic costs can be saved by using the relation $C_2(w,v) = - C_1(v,w)$. Due to the chosen data distribution, no additional communication is required !