5.8.2 Type-II preconditioner

Using a type-II preconditioning matrix ${\ensuremath{\color{green}{\sf C}}}^{-1} = \sum\limits_{i=1}^p A_i^T {\ensuremath{\color{green}{\sf C}}}_i^{-1} A_i$ requires two vector type conversions (Sec. 4.3.1). That preconditioner can be written as :

$$\underline{{\ensuremath{\color{red}\mathfrak{w}}}} \;=\; {\ensure... ...{P} A_j^T \underline{{\ensuremath{\color{green}{\sf r}}}}_j \right) \enspace .$$ (5.11)

Matrix  $\color{green}{\sf K}$ is not accumulated and therefore the submatrices  ${\ensuremath{\color{green}{\sf K}}}_i$ represent a 2 $^{\text{nd}}$ order pdes in $\Omega _i$ with homogeneous Neumann boundary conditions on the inner boundaries (interfaces) $\partial\Omega_i\setminus\partial\Omega$. This leads in the case of the Laplace operator to singular submatrices  ${\ensuremath{\color{green}{\sf K}}}_i$.

On the other hand we may choose  ${\ensuremath{\color{green}{\sf C}}}_i$ in a different way. One opportunity consists in the accumulation of matrix  ${\ensuremath{\color{green}{\sf K}}}$ and the setting ${\ensuremath{\color{green}{\sf C}}}_i = {\ensuremath{\color{red}\mathfrak{K}}}_i$. The local preconditioner  ${\ensuremath{\color{green}{\sf C}}}_i$ represents a 2 $^{\text{nd}}$ order pde in  $\widetilde{\Omega}_i$ with homogeneous Dirichlet boundary conditions. Here, $\widetilde{\Omega}_i$ denotes $\Omega _i$ extended by the next layer of surrounding elements. Thus, this preconditioner is a overlapping one, for further details see [SBG96] or the patch-by-patch method.