7.3.3 Parallelization using split boxes and distributed elements

We assume a distribution of the domain along the boundaries of the finite elements. According to the results on the distribution of boxes and elements from section 7.2.4 and 4.1.3 the choice of $\underline{{\ensuremath{\color{red}\mathfrak{u}}}}$ as accumulated vector and , , and ${\ensuremath{\color{green}{\sf B}}}$ as distributed vectors and matrices seems to be straight forward.

Remarks :

• Excluding the cases of time dependent meshes or basis functions, the mass matrix  is time independent so that the accumulation to  ${\ensuremath{\color{red}\mathfrak{M}}}$ can be done in the initialization step.
• If additionally also the elasticity matrix  is time independent then the whole accumulation to ${\ensuremath{\color{red}\mathfrak{K}}}$ can be moved into the initialization.
• If the mass matrix  ${\ensuremath{\color{red}\mathfrak{M}}}$ is a diagonal matrix then the solving of the proper system of equations consists only in a multiplication with the inverse of these accumulated diagonal entries.
• In case that ${\ensuremath{\color{red}\mathfrak{M}}}$ does not posses a diagonal pattern a lumping will produce a diagonal matrix  $\widetilde{{\ensuremath{\color{red}\mathfrak{M}}}} \makebox[0pt]{}$.
• For the application of iterative methods from chapter 5 in the solution process of system 3c) there is no accumulation of the matrix ${\ensuremath{\color{green}{\sf K}}}$ necessary. Then the solution of
  requires communication.
• If $M$ is not a diagonal matrix then is no longer an M-Matrix, i.e., we cannot apply all iterative methods for that matrix. Sometime a modified version of the classic method can be used.

Figure 7.6: Dual mesh and domain decomposition