<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
the	O
Schur	B-Algorithm
complement	I-Algorithm
method	I-Algorithm
,	O
named	O
after	O
Issai	O
Schur	O
,	O
is	O
the	O
basic	O
and	O
the	O
earliest	O
version	O
of	O
non-overlapping	O
domain	B-Algorithm
decomposition	I-Algorithm
method	I-Algorithm
,	O
also	O
called	O
iterative	B-Algorithm
substructuring	I-Algorithm
.	O
</s>
<s>
A	O
finite	B-Application
element	I-Application
problem	I-Application
is	O
split	O
into	O
non-overlapping	O
subdomains	O
,	O
and	O
the	O
unknowns	O
in	O
the	O
interiors	O
of	O
the	O
subdomains	O
are	O
eliminated	O
.	O
</s>
<s>
The	O
remaining	O
Schur	O
complement	O
system	O
on	O
the	O
unknowns	O
associated	O
with	O
subdomain	O
interfaces	O
is	O
solved	O
by	O
the	O
conjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
When	O
we	O
discretize	B-Algorithm
this	O
problem	O
we	O
get	O
an	O
N-dimensional	O
linear	O
system	O
AU	O
=	O
F	O
.	O
The	O
Schur	B-Algorithm
complement	I-Algorithm
method	I-Algorithm
splits	O
up	O
the	O
linear	O
system	O
into	O
sub-problems	O
.	O
</s>
<s>
The	O
multiplication	O
of	O
a	O
vector	O
by	O
the	O
Schur	O
complement	O
is	O
a	O
discrete	B-Algorithm
version	O
of	O
the	O
Poincaré	B-Algorithm
–	I-Algorithm
Steklov	I-Algorithm
operator	I-Algorithm
,	O
also	O
called	O
the	O
Dirichlet	B-Algorithm
to	I-Algorithm
Neumann	I-Algorithm
mapping	I-Algorithm
.	O
</s>
<s>
Second	O
,	O
passing	O
to	O
the	O
Schur	O
complement	O
reduces	O
condition	B-Algorithm
number	I-Algorithm
and	O
thus	O
tends	O
to	O
decrease	O
the	O
number	O
of	O
iterations	O
.	O
</s>
<s>
For	O
second-order	O
problems	O
,	O
such	O
as	O
the	O
Laplace	O
equation	O
or	O
linear	O
elasticity	O
,	O
the	O
matrix	O
of	O
the	O
system	O
has	O
condition	B-Algorithm
number	I-Algorithm
of	O
the	O
order	O
1/h2	O
,	O
where	O
h	O
is	O
the	O
characteristic	O
element	O
size	O
.	O
</s>
<s>
The	O
Schur	O
complement	O
,	O
however	O
,	O
has	O
condition	B-Algorithm
number	I-Algorithm
only	O
of	O
the	O
order1/h	O
.	O
</s>
<s>
For	O
performances	O
,	O
the	O
Schur	B-Algorithm
complement	I-Algorithm
method	I-Algorithm
is	O
combined	O
with	O
preconditioning	O
,	O
at	O
least	O
a	O
diagonal	O
preconditioner	O
.	O
</s>
<s>
The	O
Neumann	O
–	O
Neumann	O
method	O
and	O
the	O
Neumann	O
–	O
Dirichlet	O
method	O
are	O
the	O
Schur	B-Algorithm
complement	I-Algorithm
method	I-Algorithm
with	O
particular	O
kinds	O
of	O
preconditioners	O
.	O
</s>
