matrix is Symmetric Positive De¬nite, then so is . As a result, is also Symmetric v
Positive De¬nite in this case.
Although simple to prove, the above properties are nonetheless important. They are
summarized in the following proposition.
x¤ …¨ r£
£¦ ™#
£¨ ¡¨
¢ #%
˜
Let be a nonsingular matrix partitioned as in (13.2) and such
that the submatrix is nonsingular and let be the restriction operator onto the interface ¥#
¤
variables, i.e, the linear operator de¬ned by
¥

¤
# “
D h
¦
Then the following properties are true.
¡ £
The Schur complement matrix is nonsingular.
¡
˜ £
If is SPD, then so is .
¡
2˜¤¦#
c
bcv
£
For any , .
¡
e
Q ¤
D
v
The ¬rst property indicates that a method that uses the above block Gaussian elimi £
nation algorithm is feasible since is nonsingular. A consequence of the second property
˜
is that when is positive de¬nite, an algorithm such as the Conjugate Gradient algorithm
can be used to solve the reduced system (13.4). Finally, the third property establishes a £
relation which may allow preconditioners for to be de¬ned based on solution techniques
˜
with the matrix .
©¡I
¨§ §© ¤ © ¤© H§¢
¦
% $&$"
# !# % 14 G3E9
7 A ) 3 ) ( 9 C
) C
) 5 I )
¤ ¤ 5
©
7 9¥¥F
F &4(
(F $
I
The partitioning used in Figure 13.3 is edgebased, meaning that a given edge in the graph
does not straddle two subdomains. If two vertices are coupled, then they must belong to the
same subdomain. From the graph theory point of view, this is perhaps less common than
vertexbased partitionings in which a vertex is not shared by two partitions (except when
domains overlap). A vertexbased partitioning is illustrated in Figure 13.5.
We will call interface edges all edges that link vertices that do not belong to the same
subdomain. In the case of overlapping, this needs clari¬cation. An overlapping edge or
vertex belongs to the same subdomain. Interface edges are only those that link a vertex
to another vertex which is not in the same subdomain already, whether in the overlapping
portion or elsewhere. Interface vertices are those vertices in a given subdomain that are
adjacent to an interface edge. For the example of the ¬gure, the interface vertices for sub
domain one (bottom, left subsquare) are the vertices labeled 10 to 16. The matrix shown
at the bottom of Figure 13.5 differs from the one of Figure 13.4, because here the inter
face nodes are not relabeled the last in the global labeling as was done in Figure 13.3.
Instead, the interface nodes are labeled as the last nodes in each subdomain. The number
of interface nodes is about twice that of the edgebased partitioning.
™ —t
7 u pU¢ 7 p z7 yw— y w u
{ {˜p p { p
˜
¡&
“£§
¢
! hC §¤ !
¤ …S !
§
¥
¤ ¤
43
41 42 44
39
37 38 40
33 34 35 36