<s>
The	O
finite	B-Application
element	I-Application
method	I-Application
(	O
FEM	O
)	O
is	O
a	O
popular	O
method	O
for	O
numerically	B-General_Concept
solving	O
differential	O
equations	O
arising	O
in	O
engineering	O
and	O
mathematical	O
modeling	O
.	O
</s>
<s>
Typical	O
problem	O
areas	O
of	O
interest	O
include	O
the	O
traditional	O
fields	O
of	O
structural	B-Algorithm
analysis	I-Algorithm
,	O
heat	O
transfer	O
,	O
fluid	O
flow	O
,	O
mass	O
transport	O
,	O
and	O
electromagnetic	O
potential	O
.	O
</s>
<s>
The	O
FEM	O
is	O
a	O
general	O
numerical	B-General_Concept
method	I-General_Concept
for	O
solving	O
partial	O
differential	O
equations	O
in	O
two	O
or	O
three	O
space	O
variables	O
(	O
i.e.	O
,	O
some	O
boundary	O
value	O
problems	O
)	O
.	O
</s>
<s>
To	O
solve	O
a	O
problem	O
,	O
the	O
FEM	O
subdivides	O
a	O
large	O
system	O
into	O
smaller	O
,	O
simpler	O
parts	O
that	O
are	O
called	O
finite	B-Application
elements	I-Application
.	O
</s>
<s>
This	O
is	O
achieved	O
by	O
a	O
particular	O
space	O
discretization	B-Algorithm
in	O
the	O
space	O
dimensions	O
,	O
which	O
is	O
implemented	O
by	O
the	O
construction	O
of	O
a	O
mesh	B-Algorithm
of	O
the	O
object	O
:	O
the	O
numerical	O
domain	O
for	O
the	O
solution	O
,	O
which	O
has	O
a	O
finite	O
number	O
of	O
points	O
.	O
</s>
<s>
The	O
finite	B-Application
element	I-Application
method	I-Application
formulation	O
of	O
a	O
boundary	O
value	O
problem	O
finally	O
results	O
in	O
a	O
system	O
of	O
algebraic	O
equations	O
.	O
</s>
<s>
The	O
simple	O
equations	O
that	O
model	O
these	O
finite	B-Application
elements	I-Application
are	O
then	O
assembled	O
into	O
a	O
larger	O
system	O
of	O
equations	O
that	O
models	O
the	O
entire	O
problem	O
.	O
</s>
<s>
The	O
FEM	O
then	O
approximates	O
a	O
solution	O
by	O
minimizing	O
an	O
associated	O
error	O
function	O
via	O
the	O
calculus	B-Algorithm
of	I-Algorithm
variations	I-Algorithm
.	O
</s>
<s>
Studying	O
or	O
analyzing	O
a	O
phenomenon	O
with	O
FEM	O
is	O
often	O
referred	O
to	O
as	O
finite	B-Application
element	I-Application
analysis	I-Application
(	O
FEA	O
)	O
.	O
</s>
<s>
To	O
explain	O
the	O
approximation	O
in	O
this	O
process	O
,	O
the	O
finite	B-Application
element	I-Application
method	I-Application
is	O
commonly	O
introduced	O
as	O
a	O
special	O
case	O
of	O
Galerkin	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
process	O
,	O
in	O
mathematical	O
language	O
,	O
is	O
to	O
construct	O
an	O
integral	O
of	O
the	O
inner	O
product	O
of	O
the	O
residual	O
and	O
the	O
weight	B-Algorithm
functions	I-Algorithm
and	O
set	O
the	O
integral	O
to	O
zero	O
.	O
</s>
<s>
The	O
residual	O
is	O
the	O
error	O
caused	O
by	O
the	O
trial	O
functions	O
,	O
and	O
the	O
weight	B-Algorithm
functions	I-Algorithm
are	O
polynomial	O
approximation	O
functions	O
that	O
project	O
the	O
residual	O
.	O
</s>
<s>
Algebraic	O
equation	O
sets	O
that	O
arise	O
in	O
the	O
steady-state	O
problems	O
are	O
solved	O
using	O
numerical	O
linear	O
algebra	O
methods	O
,	O
while	O
ordinary	O
differential	O
equation	O
sets	O
that	O
arise	O
in	O
the	O
transient	O
problems	O
are	O
solved	O
by	O
numerical	O
integration	O
using	O
standard	O
techniques	O
such	O
as	O
Euler	B-Algorithm
's	I-Algorithm
method	I-Algorithm
or	O
the	B-Algorithm
Runge-Kutta	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
practical	O
application	O
of	O
FEM	O
is	O
known	O
as	O
finite	B-Application
element	I-Application
analysis	I-Application
(	O
FEA	O
)	O
.	O
</s>
<s>
It	O
includes	O
the	O
use	O
of	O
mesh	B-Architecture
generation	I-Architecture
techniques	O
for	O
dividing	O
a	O
complex	O
problem	O
into	O
small	O
elements	O
,	O
as	O
well	O
as	O
the	O
use	O
of	O
software	O
coded	O
with	O
a	O
FEM	O
algorithm	O
.	O
</s>
<s>
In	O
applying	O
FEA	O
,	O
the	O
complex	O
problem	O
is	O
usually	O
a	O
physical	O
system	O
with	O
the	O
underlying	O
physics	O
such	O
as	O
the	O
Euler	O
–	O
Bernoulli	O
beam	O
equation	O
,	O
the	O
heat	O
equation	O
,	O
or	O
the	O
Navier-Stokes	O
equations	O
expressed	O
in	O
either	O
PDE	O
or	O
integral	B-Algorithm
equations	I-Algorithm
,	O
while	O
the	O
divided	O
small	O
elements	O
of	O
the	O
complex	O
problem	O
represent	O
different	O
areas	O
in	O
the	O
physical	O
system	O
.	O
</s>
<s>
Another	O
example	O
would	O
be	O
in	O
numerical	B-General_Concept
weather	I-General_Concept
prediction	I-General_Concept
,	O
where	O
it	O
is	O
more	O
important	O
to	O
have	O
accurate	O
predictions	O
over	O
developing	O
highly	O
nonlinear	O
phenomena	O
(	O
such	O
as	O
tropical	O
cyclones	O
in	O
the	O
atmosphere	O
,	O
or	O
eddies	O
in	O
the	O
ocean	O
)	O
rather	O
than	O
relatively	O
calm	O
areas	O
.	O
</s>
<s>
A	O
clear	O
,	O
detailed	O
and	O
practical	O
presentation	O
of	O
this	O
approach	O
can	O
be	O
found	O
in	O
The	O
Finite	B-Application
Element	I-Application
Method	I-Application
for	O
Engineers	O
.	O
</s>
<s>
While	O
it	O
is	O
difficult	O
to	O
quote	O
a	O
date	O
of	O
the	O
invention	O
of	O
the	O
finite	B-Application
element	I-Application
method	I-Application
,	O
the	O
method	O
originated	O
from	O
the	O
need	O
to	O
solve	O
complex	O
elasticity	O
and	O
structural	B-Algorithm
analysis	I-Algorithm
problems	O
in	O
civil	O
and	O
aeronautical	O
engineering	O
.	O
</s>
<s>
It	O
was	O
also	O
independently	O
rediscovered	O
in	O
China	O
by	O
Feng	O
Kang	O
in	O
the	O
later	O
1950s	O
and	O
early	O
1960s	O
,	O
based	O
on	O
the	O
computations	O
of	O
dam	O
constructions	O
,	O
where	O
it	O
was	O
called	O
the	O
finite	B-Application
difference	I-Application
method	I-Application
based	I-Application
on	I-Application
variation	I-Application
principle	I-Application
.	O
</s>
<s>
Although	O
the	O
approaches	O
used	O
by	O
these	O
pioneers	O
are	O
different	O
,	O
they	O
share	O
one	O
essential	O
characteristic	O
:	O
mesh	B-Algorithm
discretization	B-Algorithm
of	O
a	O
continuous	O
domain	O
into	O
a	O
set	O
of	O
discrete	O
sub-domains	O
,	O
usually	O
called	O
elements	O
.	O
</s>
<s>
Hrennikoff	O
's	O
work	O
discretizes	B-Algorithm
the	O
domain	O
by	O
using	O
a	O
lattice	O
analogy	O
,	O
while	O
Courant	O
's	O
approach	O
divides	O
the	O
domain	O
into	O
finite	O
triangular	O
subregions	O
to	O
solve	O
second	O
order	O
elliptic	O
partial	O
differential	O
equations	O
that	O
arise	O
from	O
the	O
problem	O
of	O
torsion	O
of	O
a	O
cylinder	O
.	O
</s>
<s>
The	O
finite	B-Application
element	I-Application
method	I-Application
obtained	O
its	O
real	O
impetus	O
in	O
the	O
1960s	O
and	O
1970s	O
by	O
the	O
developments	O
of	O
J	O
.	O
H	O
.	O
Argyris	O
with	O
co-workers	O
at	O
the	O
University	O
of	O
Stuttgart	O
,	O
R	O
.	O
W	O
.	O
Clough	O
with	O
co-workers	O
at	O
UC	O
Berkeley	O
,	O
O	O
.	O
C	O
.	O
Zienkiewicz	O
with	O
co-workers	O
Ernest	O
Hinton	O
,	O
Bruce	O
Irons	O
and	O
others	O
at	O
Swansea	O
University	O
,	O
Philippe	O
G	O
.	O
Ciarlet	O
at	O
the	O
University	O
of	O
Paris	O
6	O
and	O
Richard	O
Gallagher	O
with	O
co-workers	O
at	O
Cornell	O
University	O
.	O
</s>
<s>
Further	O
impetus	O
was	O
provided	O
in	O
these	O
years	O
by	O
available	O
open	O
source	O
finite	B-Application
element	I-Application
programs	O
.	O
</s>
<s>
NASA	O
sponsored	O
the	O
original	O
version	O
of	O
NASTRAN	B-Algorithm
,	O
and	O
UC	O
Berkeley	O
made	O
the	O
finite	B-Application
element	I-Application
program	O
SAP	O
IV	O
widely	O
available	O
.	O
</s>
<s>
In	O
Norway	O
the	O
ship	O
classification	O
society	O
Det	O
Norske	O
Veritas	O
(	O
now	O
DNV	O
GL	O
)	O
developed	O
Sesam	B-Algorithm
in	O
1969	O
for	O
use	O
in	O
analysis	O
of	O
ships	O
.	O
</s>
<s>
A	O
rigorous	O
mathematical	O
basis	O
to	O
the	O
finite	B-Application
element	I-Application
method	I-Application
was	O
provided	O
in	O
1973	O
with	O
the	O
publication	O
by	O
Strang	O
and	O
Fix	O
.	O
</s>
<s>
The	O
method	O
has	O
since	O
been	O
generalized	O
for	O
the	O
numerical	B-General_Concept
modeling	I-General_Concept
of	O
physical	O
systems	O
in	O
a	O
wide	O
variety	O
of	O
engineering	O
disciplines	O
,	O
e.g.	O
,	O
electromagnetism	O
,	O
heat	O
transfer	O
,	O
and	O
fluid	O
dynamics	O
.	O
</s>
<s>
A	O
finite	B-Application
element	I-Application
method	I-Application
is	O
characterized	O
by	O
a	O
variational	B-Algorithm
formulation	I-Algorithm
,	O
a	O
discretization	B-Algorithm
strategy	O
,	O
one	O
or	O
more	O
solution	O
algorithms	O
,	O
and	O
post-processing	O
procedures	O
.	O
</s>
<s>
Examples	O
of	O
the	O
variational	B-Algorithm
formulation	I-Algorithm
are	O
the	O
Galerkin	B-Algorithm
method	I-Algorithm
,	O
the	O
discontinuous	O
Galerkin	B-Algorithm
method	I-Algorithm
,	O
mixed	O
methods	O
,	O
etc	O
.	O
</s>
<s>
A	O
discretization	B-Algorithm
strategy	O
is	O
understood	O
to	O
mean	O
a	O
clearly	O
defined	O
set	O
of	O
procedures	O
that	O
cover	O
(	O
a	O
)	O
the	O
creation	O
of	O
finite	B-Application
element	I-Application
meshes	B-Algorithm
,	O
(	O
b	O
)	O
the	O
definition	O
of	O
basis	O
function	O
on	O
reference	O
elements	O
(	O
also	O
called	O
shape	O
functions	O
)	O
and	O
(	O
c	O
)	O
the	O
mapping	O
of	O
reference	O
elements	O
onto	O
the	O
elements	O
of	O
the	O
mesh	B-Algorithm
.	O
</s>
<s>
Examples	O
of	O
discretization	B-Algorithm
strategies	O
are	O
the	O
h-version	O
,	O
p-version	B-Algorithm
,	O
hp-version	B-Algorithm
,	O
x-FEM	B-Algorithm
,	O
isogeometric	B-Algorithm
analysis	I-Algorithm
,	O
etc	O
.	O
</s>
<s>
Each	O
discretization	B-Algorithm
strategy	O
has	O
certain	O
advantages	O
and	O
disadvantages	O
.	O
</s>
<s>
A	O
reasonable	O
criterion	O
in	O
selecting	O
a	O
discretization	B-Algorithm
strategy	O
is	O
to	O
realize	O
nearly	O
optimal	O
performance	O
for	O
the	O
broadest	O
set	O
of	O
mathematical	O
models	O
in	O
a	O
particular	O
model	O
class	O
.	O
</s>
<s>
Various	O
numerical	B-General_Concept
solution	I-General_Concept
algorithms	O
can	O
be	O
classified	O
into	O
two	O
broad	O
categories	O
;	O
direct	O
and	O
iterative	O
solvers	O
.	O
</s>
<s>
These	O
algorithms	O
are	O
designed	O
to	O
exploit	O
the	O
sparsity	B-Algorithm
of	O
matrices	O
that	O
depend	O
on	O
the	O
choices	O
of	O
variational	B-Algorithm
formulation	I-Algorithm
and	O
discretization	B-Algorithm
strategy	O
.	O
</s>
<s>
Postprocessing	O
procedures	O
are	O
designed	O
for	O
the	O
extraction	O
of	O
the	O
data	O
of	O
interest	O
from	O
a	O
finite	B-Application
element	I-Application
solution	O
.	O
</s>
<s>
When	O
the	O
errors	O
of	O
approximation	O
are	O
larger	O
than	O
what	O
is	O
considered	O
acceptable	O
then	O
the	O
discretization	B-Algorithm
has	O
to	O
be	O
changed	O
either	O
by	O
an	O
automated	O
adaptive	O
process	O
or	O
by	O
the	O
action	O
of	O
the	O
analyst	O
.	O
</s>
<s>
There	O
are	O
some	O
very	O
efficient	O
postprocessors	O
that	O
provide	O
for	O
the	O
realization	O
of	O
superconvergence	B-Algorithm
.	O
</s>
<s>
The	O
following	O
two	O
problems	O
demonstrate	O
the	O
finite	B-Application
element	I-Application
method	I-Application
.	O
</s>
<s>
where	O
is	O
a	O
connected	O
open	O
region	O
in	O
the	O
plane	O
whose	O
boundary	O
is	O
nice	O
(	O
e.g.	O
,	O
a	O
smooth	O
manifold	O
or	O
a	O
polygon	B-General_Concept
)	O
,	O
and	O
and	O
denote	O
the	O
second	O
derivatives	O
with	O
respect	O
to	O
and	O
,	O
respectively	O
.	O
</s>
<s>
The	O
problem	O
P1	O
can	O
be	O
solved	O
directly	O
by	O
computing	O
antiderivatives	B-Algorithm
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
we	O
will	O
develop	O
the	O
finite	B-Application
element	I-Application
method	I-Application
for	O
P1	O
and	O
outline	O
its	O
generalization	O
to	O
P2	O
.	O
</s>
<s>
The	O
second	O
step	O
is	O
the	O
discretization	B-Algorithm
,	O
where	O
the	O
weak	O
form	O
is	O
discretized	B-Algorithm
in	O
a	O
finite-dimensional	O
space	O
.	O
</s>
<s>
The	O
first	O
step	O
is	O
to	O
convert	O
P1	O
and	O
P2	O
into	O
their	O
equivalent	O
weak	B-Algorithm
formulations	I-Algorithm
.	O
</s>
<s>
The	O
proof	O
is	O
easier	O
for	O
twice	O
continuously	O
differentiable	O
(	O
mean	B-Algorithm
value	I-Algorithm
theorem	I-Algorithm
)	O
,	O
but	O
may	O
be	O
proved	O
in	O
a	O
distributional	O
sense	O
as	O
well	O
.	O
</s>
<s>
P1	O
and	O
P2	O
are	O
ready	O
to	O
be	O
discretized	B-Algorithm
which	O
leads	O
to	O
a	O
common	O
sub-problem	O
(	O
3	O
)	O
.	O
</s>
<s>
There	O
are	O
many	O
possible	O
choices	O
for	O
(	O
one	O
possibility	O
leads	O
to	O
the	O
spectral	B-Algorithm
method	I-Algorithm
)	O
.	O
</s>
<s>
However	O
,	O
for	O
the	O
finite	B-Application
element	I-Application
method	I-Application
we	O
take	O
to	O
be	O
a	O
space	O
of	O
piecewise	O
polynomial	O
functions	O
.	O
</s>
<s>
In	O
the	O
figure	O
on	O
the	O
right	O
,	O
we	O
have	O
illustrated	O
a	O
triangulation	B-Algorithm
of	O
a	O
15	O
sided	O
polygonal	O
region	O
in	O
the	O
plane	O
(	O
below	O
)	O
,	O
and	O
a	O
piecewise	O
linear	O
function	O
(	O
above	O
,	O
in	O
color	O
)	O
of	O
this	O
polygon	B-General_Concept
which	O
is	O
linear	O
on	O
each	O
triangle	O
of	O
the	O
triangulation	B-Algorithm
;	O
the	O
space	O
would	O
consist	O
of	O
functions	O
that	O
are	O
linear	O
on	O
each	O
triangle	O
of	O
the	O
chosen	O
triangulation	B-Algorithm
.	O
</s>
<s>
One	O
hopes	O
that	O
as	O
the	O
underlying	O
triangular	O
mesh	B-Algorithm
becomes	O
finer	O
and	O
finer	O
,	O
the	O
solution	O
of	O
the	O
discrete	O
problem	O
(	O
3	O
)	O
will	O
in	O
some	O
sense	O
converge	O
to	O
the	O
solution	O
of	O
the	O
original	O
boundary	O
value	O
problem	O
P2	O
.	O
</s>
<s>
To	O
measure	O
this	O
mesh	B-Algorithm
fineness	O
,	O
the	O
triangulation	B-Algorithm
is	O
indexed	O
by	O
a	O
real-valued	O
parameter	O
which	O
one	O
takes	O
to	O
be	O
very	O
small	O
.	O
</s>
<s>
This	O
parameter	O
will	O
be	O
related	O
to	O
the	O
size	O
of	O
the	O
largest	O
or	O
average	O
triangle	O
in	O
the	O
triangulation	B-Algorithm
.	O
</s>
<s>
As	O
we	O
refine	O
the	O
triangulation	B-Algorithm
,	O
the	O
space	O
of	O
piecewise	O
linear	O
functions	O
must	O
also	O
change	O
with	O
.	O
</s>
<s>
To	O
complete	O
the	O
discretization	B-Algorithm
,	O
we	O
must	O
select	O
a	O
basis	O
of	O
.	O
</s>
<s>
For	O
the	O
two-dimensional	O
case	O
,	O
we	O
choose	O
again	O
one	O
basis	O
function	O
per	O
vertex	O
of	O
the	O
triangulation	B-Algorithm
of	O
the	O
planar	O
region	O
.	O
</s>
<s>
Depending	O
on	O
the	O
author	O
,	O
the	O
word	O
"	O
element	O
"	O
in	O
the	O
"	O
finite	B-Application
element	I-Application
method	I-Application
"	O
refers	O
either	O
to	O
the	O
triangles	O
in	O
the	O
domain	O
,	O
the	O
piecewise	O
linear	O
basis	O
function	O
,	O
or	O
both	O
.	O
</s>
<s>
So	O
for	O
instance	O
,	O
an	O
author	O
interested	O
in	O
curved	O
domains	O
might	O
replace	O
the	O
triangles	O
with	O
curved	O
primitives	B-Algorithm
,	O
and	O
so	O
might	O
describe	O
the	O
elements	O
as	O
being	O
curvilinear	O
.	O
</s>
<s>
The	O
finite	B-Application
element	I-Application
method	I-Application
is	O
not	O
restricted	O
to	O
triangles	O
(	O
or	O
tetrahedra	O
in	O
3-d	O
,	O
or	O
higher-order	O
simplexes	O
in	O
multidimensional	O
spaces	O
)	O
,	O
but	O
can	O
be	O
defined	O
on	O
quadrilateral	O
subdomains	O
(	O
hexahedra	O
,	O
prisms	O
,	O
or	O
pyramids	O
in	O
3-d	O
,	O
and	O
so	O
on	O
)	O
.	O
</s>
<s>
Examples	O
of	O
methods	O
that	O
use	O
higher	O
degree	O
piecewise	O
polynomial	O
basis	O
functions	O
are	O
the	O
hp-FEM	B-Algorithm
and	O
spectral	B-Algorithm
FEM	I-Algorithm
.	O
</s>
<s>
More	O
advanced	O
implementations	O
(	O
adaptive	O
finite	B-Application
element	I-Application
methods	I-Application
)	O
utilize	O
a	O
method	O
to	O
assess	O
the	O
quality	O
of	O
the	O
results	O
(	O
based	O
on	O
error	O
estimation	O
theory	O
)	O
and	O
modify	O
the	O
mesh	B-Algorithm
during	O
the	O
solution	O
aiming	O
to	O
achieve	O
an	O
approximate	O
solution	O
within	O
some	O
bounds	O
from	O
the	O
exact	O
solution	O
of	O
the	O
continuum	O
problem	O
.	O
</s>
<s>
Mesh	B-Algorithm
adaptivity	O
may	O
utilize	O
various	O
techniques	O
,	O
the	O
most	O
popular	O
are	O
:	O
</s>
<s>
combinations	O
of	O
the	O
above	O
(	O
hp-adaptivity	B-Algorithm
)	O
.	O
</s>
<s>
(	O
The	O
matrix	O
containing	O
in	O
the	O
location	O
is	O
known	O
as	O
the	O
Gramian	B-Algorithm
matrix	I-Algorithm
.	O
)	O
</s>
<s>
Such	O
matrices	O
are	O
known	O
as	O
sparse	B-Algorithm
matrices	I-Algorithm
,	O
and	O
there	O
are	O
efficient	O
solvers	O
for	O
such	O
problems	O
(	O
much	O
more	O
efficient	O
than	O
actually	O
inverting	O
the	O
matrix	O
.	O
)	O
</s>
<s>
In	O
addition	O
,	O
is	O
symmetric	O
and	O
positive	O
definite	O
,	O
so	O
a	O
technique	O
such	O
as	O
the	O
conjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
is	O
favored	O
.	O
</s>
<s>
For	O
instance	O
,	O
MATLAB	B-Language
's	O
backslash	O
operator	O
(	O
which	O
uses	O
sparse	O
LU	O
,	O
sparse	O
Cholesky	O
,	O
and	O
other	O
factorization	O
methods	O
)	O
can	O
be	O
sufficient	O
for	O
meshes	B-Algorithm
with	O
a	O
hundred	O
thousand	O
vertices	O
.	O
</s>
<s>
The	O
matrix	O
is	O
usually	O
referred	O
to	O
as	O
the	O
stiffness	B-Algorithm
matrix	I-Algorithm
,	O
while	O
the	O
matrix	O
is	O
dubbed	O
the	O
mass	B-General_Concept
matrix	I-General_Concept
.	O
</s>
<s>
In	O
general	O
,	O
the	O
finite	B-Application
element	I-Application
method	I-Application
is	O
characterized	O
by	O
the	O
following	O
process	O
.	O
</s>
<s>
In	O
the	O
preceding	O
treatment	O
,	O
the	O
grid	O
consisted	O
of	O
triangles	O
,	O
but	O
one	O
can	O
also	O
use	O
squares	O
or	O
curvilinear	O
polygons	B-General_Concept
.	O
</s>
<s>
If	O
this	O
condition	O
is	O
not	O
satisfied	O
,	O
we	O
obtain	O
a	O
nonconforming	O
element	O
method	O
,	O
an	O
example	O
of	O
which	O
is	O
the	O
space	O
of	O
piecewise	O
linear	O
functions	O
over	O
the	O
mesh	B-Algorithm
which	O
are	O
continuous	O
at	O
each	O
edge	O
midpoint	O
.	O
</s>
<s>
Typically	O
,	O
one	O
has	O
an	O
algorithm	O
for	O
taking	O
a	O
given	O
mesh	B-Algorithm
and	O
subdividing	O
it	O
.	O
</s>
<s>
If	O
the	O
main	O
method	O
for	O
increasing	O
precision	O
is	O
to	O
subdivide	O
the	O
mesh	B-Algorithm
,	O
one	O
has	O
an	O
h-method	O
(	O
h	O
is	O
customarily	O
the	O
diameter	O
of	O
the	O
largest	O
element	O
in	O
the	O
mesh	B-Algorithm
.	O
)	O
</s>
<s>
If	O
one	O
combines	O
these	O
two	O
refinement	O
types	O
,	O
one	O
obtains	O
an	O
hp-method	O
(	O
hp-FEM	B-Algorithm
)	O
.	O
</s>
<s>
In	O
the	O
hp-FEM	B-Algorithm
,	O
the	O
polynomial	O
degrees	O
can	O
vary	O
from	O
element	O
to	O
element	O
.	O
</s>
<s>
High	O
order	O
methods	O
with	O
large	O
uniform	O
p	O
are	O
called	O
spectral	O
finite	B-Application
element	I-Application
methods	I-Application
(	O
SFEM	B-Algorithm
)	O
.	O
</s>
<s>
These	O
are	O
not	O
to	O
be	O
confused	O
with	O
spectral	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
The	O
Applied	B-Algorithm
Element	I-Algorithm
Method	I-Algorithm
or	O
AEM	O
combines	O
features	O
of	O
both	O
FEM	O
and	O
Discrete	B-Algorithm
element	I-Algorithm
method	I-Algorithm
,	O
or	O
(	O
DEM	O
)	O
.	O
</s>
<s>
The	O
generalized	O
finite	B-Application
element	I-Application
method	I-Application
(	O
GFEM	O
)	O
uses	O
local	O
spaces	O
consisting	O
of	O
functions	O
,	O
not	O
necessarily	O
polynomials	O
,	O
that	O
reflect	O
the	O
available	O
information	O
on	O
the	O
unknown	O
solution	O
and	O
thus	O
ensure	O
good	O
local	O
approximation	O
.	O
</s>
<s>
The	O
mixed	O
finite	B-Application
element	I-Application
method	I-Application
is	O
a	O
type	O
of	O
finite	B-Application
element	I-Application
method	I-Application
in	O
which	O
extra	O
independent	O
variables	O
are	O
introduced	O
as	O
nodal	O
variables	O
during	O
the	O
discretization	B-Algorithm
of	O
a	O
partial	O
differential	O
equation	O
problem	O
.	O
</s>
<s>
The	O
hp-FEM	B-Algorithm
combines	O
adaptively	O
,	O
elements	O
with	O
variable	O
size	O
h	O
and	O
polynomial	O
degree	O
p	O
in	O
order	O
to	O
achieve	O
exceptionally	O
fast	O
,	O
exponential	O
convergence	O
rates	O
.	O
</s>
<s>
The	O
extended	B-Algorithm
finite	I-Algorithm
element	I-Algorithm
method	I-Algorithm
(	O
XFEM	B-Algorithm
)	O
is	O
a	O
numerical	O
technique	O
based	O
on	O
the	O
generalized	O
finite	B-Application
element	I-Application
method	I-Application
(	O
GFEM	O
)	O
and	O
the	O
partition	O
of	O
unity	O
method	O
(	O
PUM	O
)	O
.	O
</s>
<s>
It	O
extends	O
the	O
classical	O
finite	B-Application
element	I-Application
method	I-Application
by	O
enriching	O
the	O
solution	O
space	O
for	O
solutions	O
to	O
differential	O
equations	O
with	O
discontinuous	O
functions	O
.	O
</s>
<s>
Extended	B-Algorithm
finite	I-Algorithm
element	I-Algorithm
methods	I-Algorithm
enrich	O
the	O
approximation	O
space	O
so	O
that	O
it	O
can	O
naturally	O
reproduce	O
the	O
challenging	O
feature	O
associated	O
with	O
the	O
problem	O
of	O
interest	O
:	O
the	O
discontinuity	O
,	O
singularity	O
,	O
boundary	O
layer	O
,	O
etc	O
.	O
</s>
<s>
Moreover	O
,	O
treating	O
problems	O
with	O
discontinuities	O
with	O
XFEMs	B-Algorithm
suppresses	O
the	O
need	O
to	O
mesh	B-Algorithm
and	O
re-mesh	O
the	O
discontinuity	O
surfaces	O
,	O
thus	O
alleviating	O
the	O
computational	O
costs	O
and	O
projection	O
errors	O
associated	O
with	O
conventional	O
finite	B-Application
element	I-Application
methods	I-Application
,	O
at	O
the	O
cost	O
of	O
restricting	O
the	O
discontinuities	O
to	O
mesh	B-Algorithm
edges	O
.	O
</s>
<s>
XFEM	B-Algorithm
has	O
also	O
been	O
implemented	O
in	O
codes	O
like	O
Altair	O
Radios	O
,	O
ASTER	O
,	O
Morfeo	O
,	O
and	O
Abaqus	O
.	O
</s>
<s>
It	O
is	O
increasingly	O
being	O
adopted	O
by	O
other	O
commercial	O
finite	B-Application
element	I-Application
software	O
,	O
with	O
a	O
few	O
plugins	O
and	O
actual	O
core	O
implementations	O
available	O
(	O
ANSYS	O
,	O
SAMCEF	O
,	O
OOFELIE	O
,	O
etc	O
.	O
</s>
<s>
The	O
introduction	O
of	O
the	O
scaled	O
boundary	O
finite	B-Application
element	I-Application
method	I-Application
(	O
SBFEM	O
)	O
came	O
from	O
Song	O
and	O
Wolf	O
(	O
1997	O
)	O
.	O
</s>
<s>
The	O
SBFEM	O
has	O
been	O
one	O
of	O
the	O
most	O
profitable	O
contributions	O
in	O
the	O
area	O
of	O
numerical	B-General_Concept
analysis	I-General_Concept
of	O
fracture	O
mechanics	O
problems	O
.	O
</s>
<s>
It	O
is	O
a	O
semi-analytical	O
fundamental-solutionless	O
method	O
which	O
combines	O
the	O
advantages	O
of	O
both	O
the	O
finite	B-Application
element	I-Application
formulations	O
and	O
procedures	O
and	O
the	O
boundary	B-Algorithm
element	I-Algorithm
discretization	B-Algorithm
.	O
</s>
<s>
However	O
,	O
unlike	O
the	O
boundary	B-Algorithm
element	I-Algorithm
method	I-Algorithm
,	O
no	O
fundamental	O
differential	O
solution	O
is	O
required	O
.	O
</s>
<s>
The	O
S-FEM	O
,	O
Smoothed	O
Finite	B-Application
Element	I-Application
Methods	I-Application
,	O
is	O
a	O
particular	O
class	O
of	O
numerical	O
simulation	O
algorithms	O
for	O
the	O
simulation	O
of	O
physical	O
phenomena	O
.	O
</s>
<s>
It	O
was	O
developed	O
by	O
combining	O
meshfree	B-Algorithm
methods	I-Algorithm
with	O
the	O
finite	B-Application
element	I-Application
method	I-Application
.	O
</s>
<s>
Spectral	B-Algorithm
element	I-Algorithm
methods	I-Algorithm
combine	O
the	O
geometric	O
flexibility	O
of	O
finite	B-Application
elements	I-Application
and	O
the	O
acute	O
accuracy	O
of	O
spectral	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
Spectral	B-Algorithm
methods	I-Algorithm
are	O
the	O
approximate	O
solution	O
of	O
weak	O
form	O
partial	O
equations	O
that	O
are	O
based	O
on	O
high-order	O
Lagrangian	O
interpolants	O
and	O
used	O
only	O
with	O
certain	O
quadrature	O
rules	O
.	O
</s>
<s>
Loubignac	B-Algorithm
iteration	I-Algorithm
is	O
an	O
iterative	O
method	O
in	O
finite	B-Application
element	I-Application
methods	I-Application
.	O
</s>
<s>
Crystal	O
plasticity	O
finite	B-Application
element	I-Application
method	I-Application
(	O
CPFEM	O
)	O
is	O
an	O
advanced	O
numerical	O
tool	O
developed	O
by	O
Franz	O
Roters	O
.	O
</s>
<s>
(	O
2013	O
)	O
as	O
an	O
extension	O
of	O
mimetic	B-Algorithm
finite	B-Algorithm
difference	I-Algorithm
(	O
MFD	O
)	O
methods	O
,	O
is	O
a	O
generalisation	O
of	O
the	O
standard	O
finite	B-Application
element	I-Application
method	I-Application
for	O
arbitrary	O
element	O
geometries	O
.	O
</s>
<s>
This	O
allows	O
admission	O
of	O
general	O
polygons	B-General_Concept
(	O
or	O
polyhedra	O
in	O
3D	O
)	O
that	O
are	O
highly	O
irregular	O
and	O
non-convex	O
in	O
shape	O
.	O
</s>
<s>
Some	O
types	O
of	O
finite	B-Application
element	I-Application
methods	I-Application
(	O
conforming	O
,	O
nonconforming	O
,	O
mixed	O
finite	B-Application
element	I-Application
methods	I-Application
)	O
are	O
particular	O
cases	O
of	O
the	O
gradient	B-Algorithm
discretization	I-Algorithm
method	I-Algorithm
(	O
GDM	O
)	O
.	O
</s>
<s>
The	O
finite	B-Algorithm
difference	I-Algorithm
method	I-Algorithm
(	O
FDM	O
)	O
is	O
an	O
alternative	O
way	O
of	O
approximating	O
solutions	O
of	O
PDEs	O
.	O
</s>
<s>
FEM	O
generally	O
allows	O
for	O
more	O
flexible	O
mesh	B-Algorithm
adaptivity	O
than	O
FDM	O
.	O
</s>
<s>
E.g.	O
,	O
first-order	O
FEM	O
is	O
identical	O
to	O
FDM	O
for	O
Poisson	O
's	O
equation	O
,	O
if	O
the	O
problem	O
is	O
discretized	B-Algorithm
by	O
a	O
regular	O
rectangular	O
mesh	B-Algorithm
with	O
each	O
rectangle	O
divided	O
into	O
two	O
triangles	O
.	O
</s>
<s>
There	O
are	O
reasons	O
to	O
consider	O
the	O
mathematical	O
foundation	O
of	O
the	O
finite	B-Application
element	I-Application
approximation	O
more	O
sound	O
,	O
for	O
instance	O
,	O
because	O
the	O
quality	O
of	O
the	O
approximation	O
between	O
grid	O
points	O
is	O
poor	O
in	O
FDM	O
.	O
</s>
<s>
solving	O
for	O
deformation	O
and	O
stresses	O
in	O
solid	O
bodies	O
or	O
dynamics	O
of	O
structures	O
)	O
while	O
computational	O
fluid	O
dynamics	O
(	O
CFD	O
)	O
tend	O
to	O
use	O
FDM	O
or	O
other	O
methods	O
like	O
finite	B-Algorithm
volume	I-Algorithm
method	I-Algorithm
(	O
FVM	O
)	O
.	O
</s>
<s>
CFD	O
problems	O
usually	O
require	O
discretization	B-Algorithm
of	O
the	O
problem	O
into	O
a	O
large	O
number	O
of	O
cells/gridpoints	O
(	O
millions	O
and	O
more	O
)	O
,	O
therefore	O
the	O
cost	O
of	O
the	O
solution	O
favors	O
simpler	O
,	O
lower-order	O
approximation	O
within	O
each	O
cell	O
.	O
</s>
<s>
This	O
is	O
especially	O
true	O
for	O
'	O
external	O
flow	O
 '	O
problems	O
,	O
like	O
airflow	O
around	O
the	O
car	O
or	O
airplane	O
,	O
or	O
weather	B-General_Concept
simulation	I-General_Concept
.	O
</s>
<s>
The	O
mesh	B-Algorithm
is	O
an	O
integral	O
part	O
of	O
the	O
model	O
and	O
it	O
must	O
be	O
controlled	O
carefully	O
to	O
give	O
the	O
best	O
results	O
.	O
</s>
<s>
Generally	O
the	O
higher	O
the	O
number	O
of	O
elements	O
in	O
a	O
mesh	B-Algorithm
,	O
the	O
more	O
accurate	O
the	O
solution	O
of	O
the	O
discretized	B-Algorithm
problem	O
.	O
</s>
<s>
However	O
,	O
there	O
is	O
a	O
value	O
at	O
which	O
the	O
results	O
converge	O
and	O
further	O
mesh	B-Algorithm
refinement	O
does	O
not	O
increase	O
accuracy	O
.	O
</s>
<s>
In	O
the	O
1990s	O
FEM	O
was	O
proposed	O
for	O
use	O
in	O
stochastic	O
modelling	O
for	O
numerically	B-General_Concept
solving	O
probability	O
models	O
and	O
later	O
for	O
reliability	O
assessment	O
.	O
</s>
