4.3.1 Non-overlapping elements

Let us assume that our domain $\Omega$ is split into 4 subdomains each of them was discretized by linear triangular finite elements, see Fig. 4.5.

Figure 4.5: Non-overlapping elements.

We distinguish between 3 sets of nodes denoted by the appropriate subscripts :

''I''

nodes located in the interior of a domain [ $N_I := \sum_{i=1}^{P} N_{I,i}$],

''E''

nodes located in the interior of an interface edge [ $N_E := \sum_{j=1}^{n_e} N_{E,j}$],

''V''

cross points (vertices), i.e., nodes at start and end of an interface edge [$N_V$].

The two latter sets are often combined as coupling nodes with the subscript ''C'' [ $N_C=N_V+N_E$]. The total number of nodes is $N\:= N_V+N_E+N_I = \sum_{i=1}^P N_i$ .

For simplification purposes we number first the cross points then the edge nodes and at last the inner nodes. The nodes of one edge possess a sequential numbering, the same holds for the inner nodes of each subdomain so that all vectors and matrices have a block structure like

$$\underline{v}^T \;=\; ( \underline{v}_V,\underline{v}_{E,1},\ldot... ...rline{v}_{E,n_e}, \underline{v}_{I,1},\ldots,\underline{v}_{I,P} )^T \enspace.$$

According to the mapping of nodes to the $P$ subdomains $\overline{\Omega} \makebox[0pt]{}_i$ $(i=1,2,\ldots,P)$, all entries of matrices and vectors are distributed on the appropriate process  $\ensuremath{\mathbb{P}}_i$. The coincidence matrices $A_i$ ( $i=\overline{1,P} \makebox[0pt]{}$) represent this mapping. In detail, the $N \times N_i$ matrix $A_i$ is a boolean matrix which maps the global vector  $\underline{v}$ onto the local vector  $\underline{v}_i$.
Properties of A :

- Entries for inner nodes appear exactly once per row an column.
- Entries for coupling nodes appear once in $A_i$ if this node belongs to process  $\ensuremath{\mathbb{P}}_i$.

Now, we define 2 types of vectors - the accumulated vector (type I) and the distributed vector (type-II) :

Type I	:	$\underline{{\ensuremath{\color{red}\mathfrak{u}}}}\;$und$\;\underline{{\ensuremath{\color{red}\mathfrak{w}}}}$ are stored in process  $\ensuremath{\mathbb{P}}_i\;$ ( $\;\hat{=}\;\overline{\Omega} \makebox[0pt]{}_i\;$) in the ways $\;\underline{{\ensuremath{\color{red}\mathfrak{u}}}}_i=A_i\underline{{\ensuremath{\color{red}\mathfrak{u}}}}\;$ and $\;\underline{{\ensuremath{\color{red}\mathfrak{w}}}}_i=A_i\underline{{\ensuremath{\color{red}\mathfrak{w}}}}\;$, i.e., each process  $\ensuremath{\mathbb{P}}_i$ owns the full values of that vector.
Type II	:	$\underline{{\ensuremath{\color{green}{\sf r}}}},\underline{{\ensuremath{\color{green}{\sf f}}}}\;$ are stored in the way $\;\underline{{\ensuremath{\color{green}{\sf r}}}}_i,\underline{{\ensuremath{\color{green}{\sf f}}}}_i\;$ in $\ensuremath{\mathbb{P}}_i$ such that $\underline{{\ensuremath{\color{green}{\sf r}}}} = \sum\limits_{i=1}^p A_i^T\underline{{\ensuremath{\color{green}{\sf r}}}}_i$ holds, i.e., each node on the interface ( $\underline{{\ensuremath{\color{green}{\sf r}}}}_{C,i}$) owns only a its contribution to the full values of that vector.

Matrix $K$ is stored in a distributed way, analogously to a type-II vector, and will be classified as type-II matrix :

$${\ensuremath{\color{green}{\sf K}}} = \sum\limits_{i=1}^p A_i^T {\ensuremath{\color{green}{\sf K}}}_iA_i \enspace ,$$ (4.1)

with  ${\ensuremath{\color{green}{\sf K}}}_i$ denoting the stiffness matrix belonging to subdomain  $\overline{\Omega} \makebox[0pt]{}_i$ (to be more exact: the support of the f.e. test functions is restricted to  $\overline{\Omega} \makebox[0pt]{}_i$). If one thinks of  $\overline{\Omega} \makebox[0pt]{}_i$ as one large finite element then the distributed storing of the matrix is equivalent to the presentation of that element matrix previously to the f.e. accumulation.
The special numbering of nodes implies the following block representation of equation  ${\ensuremath{\color{green}{\sf K}}}\cdot\underline{{\ensuremath{\color{red}\mathfrak{u}}}} = \underline{{\ensuremath{\color{green}{\sf f}}}}$ : $$\begin{pmatrix}{\ensuremath{\color{green}{\sf K}}}_{V} & {\ens... ...e{{\ensuremath{\color{green}{\sf f}}}}_I \end{pmatrix} \enspace .$$
Therein, ${\ensuremath{\color{green}{\sf K}}}_I$ is a block diagonal matrix with entries  ${\ensuremath{\color{green}{\sf K}}}_{I,i}$ - similar block structures are valid for ${\ensuremath{\color{green}{\sf K}}}_{IC}$, ${\ensuremath{\color{green}{\sf K}}}_{CI}$, ${\ensuremath{\color{green}{\sf K}}}_{IV}$, ${\ensuremath{\color{green}{\sf K}}}_{VI}$.

If we really perform the global accumulation of matrix  ${\ensuremath{\color{green}{\sf K}}}$ then the result is a type-I matrix  ${\ensuremath{\color{red}\mathfrak{M}}}$ and we can write

$${\ensuremath{\color{red}\mathfrak{M}}}_i \;:=\; A_i {\ensuremath{\color{red}\mathfrak{M}}} A_i^T \enspace .$$ (4.2)

Although  ${\ensuremath{\color{green}{\sf K}}}\equiv{\ensuremath{\color{red}\mathfrak{M}}}$ holds, we have to distinguish between both representations because of the different local storing ( ${\ensuremath{\color{green}{\sf K}}}_i\not\equiv{\ensuremath{\color{red}\mathfrak{M}}}_i$) !

The diagonal matrix

$$\boxed{ R \;=\; \sum\limits_{i=1}^p A_i^T A_i }$$ (4.3)

contains for each node the number of subdomains it belongs to (priority of a node), e.g., in Fig. 4.5 $R^{[k]}:=4$, $R^{[n]}=R^{[m]}=R^{[p]}=2$, $R^{[q]}=1$.

Declaration : We will use subscripts and superscripts in the remaining section in the following way : ${\ensuremath{\color{green}{\sf v}}}_{C,i}^{[n]}$ denotes the $n^{\text{th}}$ component (in local or global numbering) of vector  $\underline{{\ensuremath{\color{green}{\sf r}}}}$ stored in process  $\ensuremath{\mathbb{P}}_i$. Subscript ''$C$'' indicates a subvector belonging to the interface. A similar notation is used for matrices.