5.4.4.2 Variant 2 : Gauß-Seidel-$\omega$-Jacobi

The large amount of communication steps of variant 1 was caused by the coupling of unknowns on the interface in the same iteration. By using only the old $(k-1)^{\text{th}}$ iterates one can omit this coupling and also the requirements on the mesh.
Between the blocks ''$V,E,I$'' and inside the subdomains (block ''$I$'') a Gauß-Seidel iteration will be performed. Due to the partial use of the Jacobi iteration, the convergence rate is worse as in a Gauß-Seidel iteration.

Operation  $\underline{{\ensuremath{\color{red}\mathfrak{d}}}} \circledast \underline{{\ensuremath{\color{red}\mathfrak{w}}}}$ in Alg. 5.10 denotes the component wise multiplication of two vectors. We took additionally into account that in the interior of the domains (''$I$'') accumulated and distributed vectors/matrices are identical.
The accumulation of cross points (''$V$'') and interface data (''$E$'') is usually performed separately so that we need exactly the same amount of communication as in the $\omega$-Jacobi iteration (Alg. 5.6) !
In case of using the iteration as a smoother with a fixed number of iterations the calculation of the inner product is no longer necessary. This saves the ALL/SMALL>_REDUCE operation in the parallel code and vectors  $\underline{{\ensuremath{\color{green}{\sf r}}}}$ and $\underline{{\ensuremath{\color{red}\mathfrak{w}}}}$ can be stored in one place.