<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
the	O
balancing	B-Algorithm
domain	I-Algorithm
decomposition	I-Algorithm
method	I-Algorithm
(	O
BDD	O
)	O
is	O
an	O
iterative	B-Algorithm
method	I-Algorithm
to	O
find	O
the	O
solution	O
of	O
a	O
symmetric	B-Algorithm
positive	O
definite	O
system	O
of	O
linear	O
algebraic	O
equations	O
arising	O
from	O
the	O
finite	B-Application
element	I-Application
method	I-Application
.	O
</s>
<s>
In	O
each	O
iteration	O
,	O
it	O
combines	O
the	O
solution	O
of	O
local	O
problems	O
on	O
non-overlapping	O
subdomains	O
with	O
a	O
coarse	B-Algorithm
problem	I-Algorithm
created	O
from	O
the	O
subdomain	O
nullspaces	B-Algorithm
.	O
</s>
<s>
BDD	O
requires	O
only	O
solution	O
of	O
subdomain	O
problems	O
rather	O
than	O
access	O
to	O
the	O
matrices	O
of	O
those	O
problems	O
,	O
so	O
it	O
is	O
applicable	O
to	O
situations	O
where	O
only	O
the	O
solution	O
operators	O
are	O
available	O
,	O
such	O
as	O
in	O
oil	O
reservoir	O
simulation	O
by	O
mixed	O
finite	B-Application
elements	I-Application
.	O
</s>
<s>
For	O
4th	O
order	O
problems	O
,	O
such	O
as	O
plate	O
bending	O
,	O
it	O
needs	O
to	O
be	O
modified	O
by	O
adding	O
to	O
the	O
coarse	B-Algorithm
problem	I-Algorithm
special	O
basis	O
functions	O
that	O
enforce	O
continuity	O
of	O
the	O
solution	O
at	O
subdomain	O
corners	O
,	O
which	O
makes	O
it	O
however	O
more	O
expensive	O
.	O
</s>
<s>
The	O
BDDC	B-Algorithm
method	O
uses	O
the	O
same	O
corner	O
basis	O
functions	O
as	O
,	O
but	O
in	O
an	O
additive	O
rather	O
than	O
multiplicative	O
fashion	O
.	O
</s>
<s>
The	O
dual	O
counterpart	O
to	O
BDD	O
is	O
FETI	B-Algorithm
,	O
which	O
enforces	O
the	O
equality	O
of	O
the	O
solution	O
between	O
the	O
subdomain	O
by	O
Lagrange	O
multipliers	O
.	O
</s>
<s>
The	O
base	O
versions	O
of	O
BDD	O
and	O
FETI	B-Algorithm
are	O
not	O
mathematically	O
equivalent	O
,	O
though	O
a	O
special	O
version	O
of	O
FETI	B-Algorithm
designed	O
to	O
be	O
robust	O
for	O
hard	O
problems	O
has	O
the	O
same	O
eigenvalues	O
and	O
thus	O
essentially	O
the	O
same	O
performance	O
as	O
BDD	O
.	O
</s>
<s>
Since	O
the	O
BDD	O
preconditioner	O
involves	O
the	O
solution	O
of	O
Neumann	O
problems	O
on	O
all	O
subdomain	O
,	O
it	O
is	O
a	O
member	O
of	O
the	O
Neumann	B-Algorithm
–	I-Algorithm
Neumann	I-Algorithm
class	I-Algorithm
of	I-Algorithm
methods	I-Algorithm
,	O
so	O
named	O
because	O
they	O
solve	O
a	O
Neumann	O
problem	O
on	O
both	O
sides	O
of	O
the	O
interface	O
between	O
subdomains	O
.	O
</s>
<s>
In	O
the	O
simplest	O
case	O
,	O
the	O
coarse	B-Algorithm
space	I-Algorithm
of	O
BDD	O
consists	O
of	O
functions	O
constant	O
on	O
each	O
subdomain	O
and	O
averaged	O
on	O
the	O
interfaces	O
.	O
</s>
<s>
More	O
generally	O
,	O
on	O
each	O
subdomain	O
,	O
the	O
coarse	B-Algorithm
space	I-Algorithm
needs	O
to	O
only	O
contain	O
the	O
nullspace	B-Algorithm
of	O
the	O
problem	O
as	O
a	O
subspace	O
.	O
</s>
