5.7.2.1 Interpolation

Investigating Fig. 5.6 and the proper interpolation resulting from the mesh refinement we can derive for the three sets of nodes $V,E,I$ the following statements :

1. The interpolation  $I_{V,q-1}^{q}$ within the set of cross points is the (potentially scaled) identity $I\enspace$.
2. The complete set of crosspoints belongs to the coarse grid and therefore they will never be interpolated by nodes belonging to $E$ or $I$
   $\Longrightarrow$ $I_{VE,q-1}^{q} = I_{VI,q-1}^{q} = 0 \enspace$.
3. The interpolation matrix on the edges breaks up into blocks
   $\Longrightarrow$ $I_{E,q-1}^{q} = \textrm{blockdiag}\{I_{E_j,q-1}^{q}\}_{i=\overline{1,\textrm{NumEdges}} \makebox[0pt]{}} \enspace$.
4. Nodes on the edges will never be interpolated by inner nodes
   $\Longrightarrow$ $I_{EI,q-1}^{q} = 0 \enspace$.
5. Obviously, also the interpolation matrix of the inner nodes ($I$) is block diagonal
   $\Longrightarrow$ $I_{I,q-1}^{q} = \textrm{blockdiag}\{I_{I_s,q-1}^{q}\}_{s=\overline{1,P} \makebox[0pt]{}} \enspace$.

Therefore, the interpolation matrix  $I_{q-1}^{q}$ possesses a block structure in which the submatrices on the upper right are equal 0. Especially, the structure of the matrix allows the application of (4.9).

$\Longrightarrow$

The interpolation matrix has to be accumulated : ${\ensuremath{\color{red}\mathfrak{I}}}_{q-1}^{q}$ .

$\Longrightarrow$

Vector  $\underline{{\ensuremath{\color{red}\mathfrak{w}}}}_{q-1}$ is accumulated.

In principle it would be also possible to define the interpolation matrix as distributed. But in that case an additional type conversion with communication has to be performed after the interpolation or in the following addition of the correction.

No communication in interpolation !