<s>
In	O
the	O
finite	B-Application
element	I-Application
method	I-Application
for	O
the	O
numerical	O
solution	O
of	O
elliptic	O
partial	O
differential	O
equations	O
,	O
the	O
stiffness	B-Algorithm
matrix	I-Algorithm
is	O
a	O
matrix	B-Architecture
that	O
represents	O
the	O
system	O
of	O
linear	O
equations	O
that	O
must	O
be	O
solved	O
in	O
order	O
to	O
ascertain	O
an	O
approximate	O
solution	O
to	O
the	O
differential	O
equation	O
.	O
</s>
<s>
To	O
discretize	B-Algorithm
this	O
equation	O
by	O
the	O
finite	B-Application
element	I-Application
method	I-Application
,	O
one	O
chooses	O
a	O
set	O
of	O
basis	O
functions	O
defined	O
on	O
which	O
also	O
vanish	O
on	O
the	O
boundary	O
.	O
</s>
<s>
The	O
stiffness	B-Algorithm
matrix	I-Algorithm
is	O
symmetric	B-Algorithm
,	O
i.e.	O
</s>
<s>
Moreover	O
,	O
it	O
is	O
a	O
strictly	O
positive-definite	B-Algorithm
matrix	I-Algorithm
,	O
so	O
that	O
the	O
system	O
always	O
has	O
a	O
unique	O
solution	O
.	O
</s>
<s>
Note	O
that	O
the	O
stiffness	B-Algorithm
matrix	I-Algorithm
will	O
be	O
different	O
depending	O
on	O
the	O
computational	O
grid	O
used	O
for	O
the	O
domain	O
and	O
what	O
type	O
of	O
finite	B-Application
element	I-Application
is	O
used	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
stiffness	B-Algorithm
matrix	I-Algorithm
when	O
piecewise	O
quadratic	O
finite	B-Application
elements	I-Application
are	O
used	O
will	O
have	O
more	O
degrees	O
of	O
freedom	O
than	O
piecewise	O
linear	O
elements	O
.	O
</s>
<s>
Determining	O
the	O
stiffness	B-Algorithm
matrix	I-Algorithm
for	O
other	O
PDEs	O
follows	O
essentially	O
the	O
same	O
procedure	O
,	O
but	O
it	O
can	O
be	O
complicated	O
by	O
the	O
choice	O
of	O
boundary	O
conditions	O
.	O
</s>
<s>
where	O
is	O
a	O
positive-definite	B-Algorithm
matrix	I-Algorithm
defined	O
for	O
each	O
point	O
in	O
the	O
domain	O
.	O
</s>
<s>
The	O
coefficients	O
are	O
still	O
found	O
by	O
solving	O
a	O
system	O
of	O
linear	O
equations	O
,	O
but	O
the	O
matrix	B-Architecture
representing	O
the	O
system	O
is	O
markedly	O
different	O
from	O
that	O
for	O
the	O
ordinary	O
Poisson	O
problem	O
.	O
</s>
<s>
In	O
order	O
to	O
implement	O
the	O
finite	B-Application
element	I-Application
method	I-Application
on	O
a	O
computer	O
,	O
one	O
must	O
first	O
choose	O
a	O
set	O
of	O
basis	O
functions	O
and	O
then	O
compute	O
the	O
integrals	O
defining	O
the	O
stiffness	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Usually	O
,	O
the	O
domain	O
is	O
discretized	B-Algorithm
by	O
some	O
form	O
of	O
mesh	B-Architecture
generation	I-Architecture
,	O
wherein	O
it	O
is	O
divided	O
into	O
non-overlapping	O
triangles	B-Algorithm
or	O
quadrilaterals	B-Algorithm
,	O
which	O
are	O
generally	O
referred	O
to	O
as	O
elements	O
.	O
</s>
<s>
The	O
element	O
stiffness	B-Algorithm
matrix	I-Algorithm
is	O
zero	O
for	O
most	O
values	O
of	O
and	O
,	O
for	O
which	O
the	O
corresponding	O
basis	O
functions	O
are	O
zero	O
within	O
.	O
</s>
<s>
The	O
full	O
stiffness	B-Algorithm
matrix	I-Algorithm
is	O
the	O
sum	O
of	O
the	O
element	O
stiffness	O
matrices	O
.	O
</s>
<s>
In	O
particular	O
,	O
for	O
basis	O
functions	O
that	O
are	O
only	O
supported	O
locally	O
,	O
the	O
stiffness	B-Algorithm
matrix	I-Algorithm
is	O
sparse	B-Algorithm
.	O
</s>
<s>
piecewise	O
linear	O
basis	O
functions	O
on	O
triangles	B-Algorithm
,	O
there	O
are	O
simple	O
formulas	O
for	O
the	O
element	O
stiffness	O
matrices	O
.	O
</s>
<s>
When	O
the	O
differential	O
equation	O
is	O
more	O
complicated	O
,	O
say	O
by	O
having	O
an	O
inhomogeneous	O
diffusion	O
coefficient	O
,	O
the	O
integral	O
defining	O
the	O
element	O
stiffness	B-Algorithm
matrix	I-Algorithm
can	O
be	O
evaluated	O
by	O
Gaussian	B-Algorithm
quadrature	I-Algorithm
.	O
</s>
<s>
The	O
condition	B-Algorithm
number	I-Algorithm
of	O
the	O
stiffness	B-Algorithm
matrix	I-Algorithm
depends	O
strongly	O
on	O
the	O
quality	O
of	O
the	O
numerical	O
grid	O
.	O
</s>
<s>
In	O
particular	O
,	O
triangles	B-Algorithm
with	O
small	O
angles	O
in	O
the	O
finite	B-Application
element	I-Application
mesh	O
induce	O
large	O
eigenvalues	O
of	O
the	O
stiffness	B-Algorithm
matrix	I-Algorithm
,	O
degrading	O
the	O
solution	O
quality	O
.	O
</s>
