7.2.3 Parallelization using distributed boxes

The distribution of complete boxes on the processors is related to the non-overlapping node distribution from section 4.3.2.
The remaining crucial point in the parallelization of (7.17) consists in the efficient calculation of the sum $C_1(w,v)w+C_2(w,v)v$ in (7.18).

One can think of the unknowns $u_j$ stored in the center of volume elements. The local contributions of the flux  $\sigma_{jl}$ in algorithm 7.8 are stored inconsistently on the edges (faces), i.e., $\sigma_{jl} \neq \sigma_{lj}$. For the parallel implementation, an overlapping distribution of $u$ and $\sigma$ with one element overlap is necessary. Denoting by  $\widetilde{\Omega} \makebox[0pt]{}_i$ the extended subdomain of $\Omega _i$ the extended solution vector can be written as $\underline{\widetilde{u} \makebox[0pt]{}}_i = \Big( \underbrace{ \left\{ u... ...mega} \makebox[0pt]{}_i\setminus\Omega_i} }_{\mathrm{external \; comp.}} \Big)$, analogously  $\widetilde{\sigma} \makebox[0pt]{}$.

Figure 7.3: Distributed boxes

Remarks :

1. Algorithm 7.10 can be used for all 3 formulas (7.19)-(7.21).
2. By using (7.19) the values $\widetilde{u} \makebox[0pt]{}_l$ are not needed so that step 1.) can be omitted.
3. The data exchange in step 3.) by using (7.20) or (7.21) is not necessary if one calculates the value $\widetilde{\sigma} \makebox[0pt]{}_{lj} := - C_2(u_j,\widetilde{u} \makebox[0pt]{}_l)$ in  $\widetilde{\Omega} \makebox[0pt]{}_j$ redundantly to $C_1(u_l,\widetilde{u} \makebox[0pt]{}_j$) in $\widetilde{\Omega} \makebox[0pt]{}_{l}$.
   
   Disadvantages:
   • More arithmetic costs (nearly doubling).
   • Performing the redundant calculations only in the boundary elements means additional alternatives in the code.