<s>
In	O
mathematics	O
,	O
the	O
discrete	B-Algorithm
exterior	I-Algorithm
calculus	I-Algorithm
(	O
DEC	O
)	O
is	O
the	O
extension	O
of	O
the	O
exterior	O
calculus	O
to	O
discrete	O
spaces	O
including	O
graphs	O
,	O
finite	B-Application
element	I-Application
meshes	I-Application
,	O
and	O
lately	O
also	O
general	O
polygonal	O
meshes	O
(	O
non-flat	O
and	O
non-convex	O
)	O
.	O
</s>
<s>
DEC	O
methods	O
have	O
proved	O
to	O
be	O
very	O
powerful	O
in	O
improving	O
and	O
analyzing	O
finite	B-Application
element	I-Application
methods	I-Application
:	O
for	O
instance	O
,	O
DEC-based	O
methods	O
allow	O
the	O
use	O
of	O
highly	O
non-uniform	O
meshes	O
to	O
obtain	O
accurate	O
results	O
.	O
</s>
<s>
One	O
could	O
think	O
of	O
differential	O
k-forms	O
as	O
linear	B-Architecture
operators	I-Architecture
that	O
act	O
on	O
k-dimensional	O
"	O
bits	O
"	O
of	O
space	O
,	O
in	O
which	O
case	O
one	O
might	O
prefer	O
to	O
use	O
the	O
bracket	O
notation	O
for	O
a	O
dual	O
pairing	O
.	O
</s>
<s>
In	O
finite	B-Application
element	I-Application
analysis	I-Application
,	O
the	O
first	O
stage	O
is	O
often	O
the	O
approximation	O
of	O
the	O
domain	O
of	O
interest	O
by	O
a	O
triangulation	O
,	O
T	O
.	O
For	O
example	O
,	O
a	O
curve	O
would	O
be	O
approximated	O
as	O
a	O
union	O
of	O
straight	O
line	O
segments	O
;	O
a	O
surface	O
would	O
be	O
approximated	O
by	O
a	O
union	O
of	O
triangles	O
,	O
whose	O
edges	O
are	O
straight	O
line	O
segments	O
,	O
which	O
themselves	O
terminate	O
in	O
points	O
.	O
</s>
<s>
A	O
k-form	O
on	O
T	O
is	O
a	O
linear	B-Architecture
operator	I-Architecture
acting	O
on	O
k-dimensional	O
subcomplexes	O
of	O
T	O
;	O
e.g.	O
,	O
a	O
0-form	O
assigns	O
values	O
to	O
points	O
,	O
and	O
extends	O
linearly	O
to	O
linear	O
combinations	O
of	O
points	O
;	O
a	O
1-form	O
assigns	O
values	O
to	O
line	O
segments	O
in	O
a	O
similarly	O
linear	O
way	O
.	O
</s>
