<s>
In	O
mathematics	O
,	O
a	O
Poincaré	B-Algorithm
–	I-Algorithm
Steklov	I-Algorithm
operator	I-Algorithm
(	O
after	O
Henri	O
Poincaré	O
and	O
Vladimir	O
Steklov	O
)	O
maps	O
the	O
values	O
of	O
one	O
boundary	O
condition	O
of	O
the	O
solution	O
of	O
an	O
elliptic	O
partial	O
differential	O
equation	O
in	O
a	O
domain	O
to	O
the	O
values	O
of	O
another	O
boundary	O
condition	O
.	O
</s>
<s>
Thus	O
,	O
a	O
Poincaré	B-Algorithm
–	I-Algorithm
Steklov	I-Algorithm
operator	I-Algorithm
encapsulates	O
the	O
boundary	O
response	O
of	O
the	O
system	O
modelled	O
by	O
the	O
partial	O
differential	O
equation	O
.	O
</s>
<s>
When	O
the	O
partial	O
differential	O
equation	O
is	O
discretized	O
,	O
for	O
example	O
by	O
finite	B-Application
elements	I-Application
or	O
finite	B-Algorithm
differences	I-Algorithm
,	O
the	O
discretization	O
of	O
the	O
Poincaré	B-Algorithm
–	I-Algorithm
Steklov	I-Algorithm
operator	I-Algorithm
is	O
the	O
Schur	O
complement	O
obtained	O
by	O
eliminating	O
all	O
degrees	O
of	O
freedom	O
inside	O
the	O
domain	O
.	O
</s>
<s>
Note	O
that	O
there	O
may	O
be	O
many	O
suitable	O
different	O
boundary	O
conditions	O
for	O
a	O
given	O
partial	O
differential	O
equation	O
and	O
the	O
direction	O
in	O
which	O
a	O
Poincaré	B-Algorithm
–	I-Algorithm
Steklov	I-Algorithm
operator	I-Algorithm
maps	O
the	O
values	O
of	O
one	O
into	O
another	O
is	O
given	O
only	O
by	O
a	O
convention	O
.	O
</s>
<s>
The	O
mapping	O
of	O
the	O
surface	O
temperature	O
to	O
the	O
surface	O
heat	O
flux	O
is	O
a	O
Poincaré	B-Algorithm
–	I-Algorithm
Steklov	I-Algorithm
operator	I-Algorithm
.	O
</s>
<s>
This	O
particular	O
Poincaré	B-Algorithm
–	I-Algorithm
Steklov	I-Algorithm
operator	I-Algorithm
is	O
called	O
the	O
Dirichlet	O
to	O
Neumann	O
(	O
DtN	O
)	O
operator	O
.	O
</s>
<s>
Mathematically	O
,	O
for	O
a	O
function	O
harmonic	O
in	O
a	O
domain	O
,	O
the	O
Dirichlet-to-Neumann	B-Algorithm
operator	I-Algorithm
maps	O
the	O
values	O
of	O
on	O
the	O
boundary	O
of	O
to	O
the	O
normal	O
derivative	O
on	O
the	O
boundary	O
of	O
.	O
</s>
<s>
This	O
Poincaré	B-Algorithm
–	I-Algorithm
Steklov	I-Algorithm
operator	I-Algorithm
is	O
at	O
the	O
foundation	O
of	O
iterative	B-Algorithm
substructuring	I-Algorithm
.	O
</s>
<s>
Calderón	O
's	O
inverse	O
boundary	O
problem	O
is	O
the	O
problem	O
of	O
finding	O
the	O
coefficient	O
of	O
a	O
divergence	O
form	O
elliptic	O
partial	O
differential	O
equation	O
from	O
its	O
Dirichlet-to-Neumann	B-Algorithm
operator	I-Algorithm
.	O
</s>
<s>
The	O
solution	O
of	O
partial	O
differential	O
equation	O
in	O
an	O
external	O
domain	O
gives	O
rise	O
to	O
a	O
Poincaré	B-Algorithm
–	I-Algorithm
Steklov	I-Algorithm
operator	I-Algorithm
that	O
brings	O
the	O
boundary	O
condition	O
from	O
infinity	O
to	O
the	O
boundary	O
.	O
</s>
<s>
One	O
example	O
is	O
the	O
Dirichlet-to-Neumann	B-Algorithm
operator	I-Algorithm
that	O
maps	O
the	O
given	O
temperature	O
on	O
the	O
boundary	O
of	O
a	O
cavity	O
in	O
infinite	O
medium	O
with	O
zero	O
temperature	O
at	O
infinity	O
to	O
the	O
heat	O
flux	O
on	O
the	O
cavity	O
boundary	O
.	O
</s>
<s>
Similarly	O
,	O
one	O
can	O
define	O
the	O
Dirichlet-to-Neumann	B-Algorithm
operator	I-Algorithm
on	O
the	O
boundary	O
of	O
a	O
sphere	O
for	O
the	O
solution	O
for	O
the	O
Helmholtz	O
equation	O
in	O
the	O
exterior	O
of	O
the	O
sphere	O
.	O
</s>
<s>
Approximations	O
of	O
this	O
operator	O
are	O
at	O
the	O
foundation	O
of	O
a	O
class	O
of	O
method	O
for	O
the	O
modeling	O
of	O
acoustic	O
scattering	O
in	O
infinite	O
medium	O
,	O
with	O
the	O
scatterer	O
enclosed	O
in	O
the	O
sphere	O
and	O
the	O
Poincaré	B-Algorithm
–	I-Algorithm
Steklov	I-Algorithm
operator	I-Algorithm
serving	O
as	O
a	O
non-reflective	O
(	O
or	O
absorbing	O
)	O
boundary	O
condition	O
.	O
</s>
<s>
The	O
Poincaré	B-Algorithm
–	I-Algorithm
Steklov	I-Algorithm
operator	I-Algorithm
is	O
defined	O
to	O
be	O
the	O
operator	O
mapping	O
the	O
time-harmonic	O
(	O
that	O
is	O
,	O
dependent	O
on	O
time	O
as	O
)	O
tangential	O
electric	O
field	O
on	O
the	O
boundary	O
of	O
a	O
region	O
to	O
the	O
equivalent	O
electric	O
current	O
on	O
its	O
boundary	O
.	O
</s>
