<s>
Numerical	B-Algorithm
methods	I-Algorithm
for	I-Algorithm
partial	I-Algorithm
differential	I-Algorithm
equations	I-Algorithm
is	O
the	O
branch	O
of	O
numerical	B-General_Concept
analysis	I-General_Concept
that	O
studies	O
the	O
numerical	B-General_Concept
solution	I-General_Concept
of	O
partial	O
differential	O
equations	O
(	O
PDEs	O
)	O
.	O
</s>
<s>
The	O
method	O
of	O
lines	O
(	O
MOL	O
,	O
NMOL	O
,	O
NUMOL	O
)	O
is	O
a	O
technique	O
for	O
solving	O
partial	O
differential	O
equations	O
(	O
PDEs	O
)	O
in	O
which	O
all	O
dimensions	O
except	O
one	O
are	O
discretized	B-Algorithm
.	O
</s>
<s>
A	O
large	O
number	O
of	O
integration	O
routines	O
have	O
been	O
developed	O
over	O
the	O
years	O
in	O
many	O
different	O
programming	O
languages	O
,	O
and	O
some	O
have	O
been	O
published	O
as	O
open	B-License
source	I-License
resources	O
.	O
</s>
<s>
The	O
method	O
of	O
lines	O
most	O
often	O
refers	O
to	O
the	O
construction	O
or	O
analysis	O
of	O
numerical	B-Algorithm
methods	I-Algorithm
for	I-Algorithm
partial	I-Algorithm
differential	I-Algorithm
equations	I-Algorithm
that	O
proceeds	O
by	O
first	O
discretizing	O
the	O
spatial	O
derivatives	O
only	O
and	O
leaving	O
the	O
time	O
variable	O
continuous	O
.	O
</s>
<s>
The	O
finite	B-Application
element	I-Application
method	I-Application
(	O
FEM	O
)	O
is	O
a	O
numerical	B-General_Concept
technique	I-General_Concept
for	O
finding	O
approximate	O
solutions	O
to	O
boundary	O
value	O
problems	O
for	O
differential	O
equations	O
.	O
</s>
<s>
It	O
uses	O
variational	B-Algorithm
methods	I-Algorithm
(	O
the	O
calculus	B-Algorithm
of	I-Algorithm
variations	I-Algorithm
)	O
to	O
minimize	O
an	O
error	O
function	O
and	O
produce	O
a	O
stable	O
solution	O
.	O
</s>
<s>
Analogous	O
to	O
the	O
idea	O
that	O
connecting	O
many	O
tiny	O
straight	O
lines	O
can	O
approximate	O
a	O
larger	O
circle	O
,	O
FEM	O
encompasses	O
all	O
the	O
methods	O
for	O
connecting	O
many	O
simple	O
element	O
equations	O
over	O
many	O
small	O
subdomains	O
,	O
named	O
finite	B-Application
elements	I-Application
,	O
to	O
approximate	O
a	O
more	O
complex	O
equation	O
over	O
a	O
larger	O
domain	B-Algorithm
.	O
</s>
<s>
The	O
gradient	O
discretization	B-Algorithm
method	I-Algorithm
(	O
GDM	O
)	O
is	O
a	O
numerical	B-General_Concept
technique	I-General_Concept
that	O
encompasses	O
a	O
few	O
standard	O
or	O
recent	O
methods	O
.	O
</s>
<s>
Core	O
properties	O
allow	O
the	O
convergence	B-Application
of	O
the	O
method	O
for	O
a	O
series	O
of	O
linear	O
and	O
nonlinear	O
problems	O
,	O
and	O
therefore	O
all	O
the	O
methods	O
that	O
enter	O
the	O
GDM	O
framework	O
(	O
conforming	O
and	O
nonconforming	O
finite	B-Application
element	I-Application
,	O
mixed	O
finite	B-Application
element	I-Application
,	O
mimetic	O
finite	O
difference	O
...	O
)	O
inherit	O
these	O
convergence	B-Application
properties	O
.	O
</s>
<s>
Similar	O
to	O
the	O
finite	B-Algorithm
difference	I-Algorithm
method	I-Algorithm
or	O
finite	B-Application
element	I-Application
method	I-Application
,	O
values	O
are	O
calculated	O
at	O
discrete	O
places	O
on	O
a	O
meshed	O
geometry	O
.	O
</s>
<s>
In	O
the	O
finite	O
volume	O
method	O
,	O
volume	O
integrals	O
in	O
a	O
partial	O
differential	O
equation	O
that	O
contain	O
a	O
divergence	B-Application
term	O
are	O
converted	O
to	O
surface	O
integrals	O
,	O
using	O
the	O
divergence	B-Application
theorem	O
.	O
</s>
<s>
Spectral	O
methods	O
are	O
techniques	O
used	O
in	O
applied	O
mathematics	O
and	O
scientific	O
computing	O
to	O
numerically	B-General_Concept
solve	O
certain	O
differential	O
equations	O
,	O
often	O
involving	O
the	O
use	O
of	O
the	O
fast	O
Fourier	O
transform	O
.	O
</s>
<s>
Spectral	O
methods	O
and	O
finite	B-Application
element	I-Application
methods	I-Application
are	O
closely	O
related	O
and	O
built	O
on	O
the	O
same	O
ideas	O
;	O
the	O
main	O
difference	O
between	O
them	O
is	O
that	O
spectral	O
methods	O
use	O
basis	O
functions	O
that	O
are	O
nonzero	O
over	O
the	O
whole	O
domain	B-Algorithm
,	O
while	O
finite	B-Application
element	I-Application
methods	I-Application
use	O
basis	O
functions	O
that	O
are	O
nonzero	O
only	O
on	O
small	O
subdomains	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
spectral	O
methods	O
take	O
on	O
a	O
global	O
approach	O
while	O
finite	B-Application
element	I-Application
methods	I-Application
use	O
a	O
local	O
approach	O
.	O
</s>
<s>
Partially	O
for	O
this	O
reason	O
,	O
spectral	O
methods	O
have	O
excellent	O
error	O
properties	O
,	O
with	O
the	O
so-called	O
"	O
exponential	O
convergence	B-Application
"	O
being	O
the	O
fastest	O
possible	O
,	O
when	O
the	O
solution	O
is	O
smooth	O
.	O
</s>
<s>
However	O
,	O
there	O
are	O
no	O
known	O
three-dimensional	O
single	O
domain	B-Algorithm
spectral	O
shock	O
capturing	O
results	O
.	O
</s>
<s>
In	O
the	O
finite	B-Application
element	I-Application
community	O
,	O
a	O
method	O
where	O
the	O
degree	O
of	O
the	O
elements	O
is	O
very	O
high	O
or	O
increases	O
as	O
the	O
grid	O
parameter	O
h	O
decreases	O
to	O
zero	O
is	O
sometimes	O
called	O
a	O
spectral	B-Algorithm
element	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Meshfree	O
methods	O
do	O
not	O
require	O
a	O
mesh	O
connecting	O
the	O
data	O
points	O
of	O
the	O
simulation	O
domain	B-Algorithm
.	O
</s>
<s>
Domain	B-Algorithm
decomposition	O
methods	O
solve	O
a	O
boundary	O
value	O
problem	O
by	O
splitting	O
it	O
into	O
smaller	O
boundary	O
value	O
problems	O
on	O
subdomains	O
and	O
iterating	O
to	O
coordinate	O
the	O
solution	O
between	O
adjacent	O
subdomains	O
.	O
</s>
<s>
A	O
coarse	B-Algorithm
problem	I-Algorithm
with	O
one	O
or	O
few	O
unknowns	O
per	O
subdomain	O
is	O
used	O
to	O
further	O
coordinate	O
the	O
solution	O
between	O
the	O
subdomains	O
globally	O
.	O
</s>
<s>
The	O
problems	O
on	O
the	O
subdomains	O
are	O
independent	O
,	O
which	O
makes	O
domain	B-Algorithm
decomposition	O
methods	O
suitable	O
for	O
parallel	B-Operating_System
computing	I-Operating_System
.	O
</s>
<s>
Domain	B-Algorithm
decomposition	O
methods	O
are	O
typically	O
used	O
as	O
preconditioners	O
for	O
Krylov	O
space	O
iterative	B-Algorithm
methods	I-Algorithm
,	O
such	O
as	O
the	O
conjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
or	O
GMRES	O
.	O
</s>
<s>
In	O
overlapping	O
domain	B-Algorithm
decomposition	O
methods	O
,	O
the	O
subdomains	O
overlap	O
by	O
more	O
than	O
the	O
interface	O
.	O
</s>
<s>
Overlapping	O
domain	B-Algorithm
decomposition	O
methods	O
include	O
the	O
Schwarz	O
alternating	O
method	O
and	O
the	O
additive	B-Algorithm
Schwarz	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Many	O
domain	B-Algorithm
decomposition	O
methods	O
can	O
be	O
written	O
and	O
analyzed	O
as	O
a	O
special	O
case	O
of	O
the	O
abstract	B-Algorithm
additive	I-Algorithm
Schwarz	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
In	O
primal	O
methods	O
,	O
such	O
as	O
Balancing	B-Algorithm
domain	I-Algorithm
decomposition	I-Algorithm
and	O
BDDC	B-Algorithm
,	O
the	O
continuity	O
of	O
the	O
solution	O
across	O
subdomain	O
interface	O
is	O
enforced	O
by	O
representing	O
the	O
value	O
of	O
the	O
solution	O
on	O
all	O
neighboring	O
subdomains	O
by	O
the	O
same	O
unknown	O
.	O
</s>
<s>
In	O
dual	O
methods	O
,	O
such	O
as	O
FETI	B-Algorithm
,	O
the	O
continuity	O
of	O
the	O
solution	O
across	O
the	O
subdomain	O
interface	O
is	O
enforced	O
by	O
Lagrange	O
multipliers	O
.	O
</s>
<s>
The	O
FETI-DP	B-Algorithm
method	O
is	O
hybrid	O
between	O
a	O
dual	O
and	O
a	O
primal	O
method	O
.	O
</s>
<s>
Non-overlapping	O
domain	B-Algorithm
decomposition	O
methods	O
are	O
also	O
called	O
iterative	O
substructuring	O
methods	O
.	O
</s>
<s>
Mortar	B-Algorithm
methods	I-Algorithm
are	O
discretization	B-Algorithm
methods	I-Algorithm
for	O
partial	O
differential	O
equations	O
,	O
which	O
use	O
separate	O
discretization	B-Algorithm
on	O
nonoverlapping	O
subdomains	O
.	O
</s>
<s>
In	O
the	O
engineering	O
practice	O
in	O
the	O
finite	B-Application
element	I-Application
method	I-Application
,	O
continuity	O
of	O
solutions	O
between	O
non-matching	O
subdomains	O
is	O
implemented	O
by	O
multiple-point	O
constraints	O
.	O
</s>
<s>
Finite	B-Application
element	I-Application
simulations	O
of	O
moderate	O
size	O
models	O
require	O
solving	O
linear	O
systems	O
with	O
millions	O
of	O
unknowns	O
.	O
</s>
<s>
Several	O
hours	O
per	O
time	O
step	O
is	O
an	O
average	O
sequential	O
run	O
time	O
,	O
therefore	O
,	O
parallel	B-Operating_System
computing	I-Operating_System
is	O
a	O
necessity	O
.	O
</s>
<s>
Domain	B-Algorithm
decomposition	O
methods	O
embody	O
large	O
potential	O
for	O
a	O
parallelization	B-Operating_System
of	O
the	O
finite	B-Application
element	I-Application
methods	I-Application
,	O
and	O
serve	O
a	O
basis	O
for	O
distributed	O
,	O
parallel	B-Operating_System
computations	I-Operating_System
.	O
</s>
<s>
Multigrid	O
(	O
MG	O
)	O
methods	O
in	O
numerical	B-General_Concept
analysis	I-General_Concept
are	O
a	O
group	O
of	O
algorithms	O
for	O
solving	O
differential	O
equations	O
using	O
a	O
hierarchy	O
of	O
discretizations	B-Algorithm
.	O
</s>
<s>
They	O
are	O
an	O
example	O
of	O
a	O
class	O
of	O
techniques	O
called	O
multiresolution	O
methods	O
,	O
very	O
useful	O
in	O
(	O
but	O
not	O
limited	O
to	O
)	O
problems	O
exhibiting	O
multiple	B-Algorithm
scales	I-Algorithm
of	O
behavior	O
.	O
</s>
<s>
For	O
example	O
,	O
many	O
basic	O
relaxation	B-Algorithm
methods	I-Algorithm
exhibit	O
different	O
rates	O
of	O
convergence	B-Application
for	O
short	O
-	O
and	O
long-wavelength	O
components	O
,	O
suggesting	O
these	O
different	O
scales	O
be	O
treated	O
differently	O
,	O
as	O
in	O
a	O
Fourier	O
analysis	O
approach	O
to	O
multigrid	O
.	O
</s>
<s>
The	O
main	O
idea	O
of	O
multigrid	O
is	O
to	O
accelerate	O
the	O
convergence	B-Application
of	O
a	O
basic	O
iterative	B-Algorithm
method	I-Algorithm
by	O
global	O
correction	O
from	O
time	O
to	O
time	O
,	O
accomplished	O
by	O
solving	O
a	O
coarse	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
This	O
principle	O
is	O
similar	O
to	O
interpolation	B-Algorithm
between	O
coarser	O
and	O
finer	O
grids	O
.	O
</s>
<s>
The	O
typical	O
application	O
for	O
multigrid	O
is	O
in	O
the	O
numerical	B-General_Concept
solution	I-General_Concept
of	O
elliptic	O
partial	O
differential	O
equations	O
in	O
two	O
or	O
more	O
dimensions	O
.	O
</s>
<s>
Multigrid	O
methods	O
can	O
be	O
applied	O
in	O
combination	O
with	O
any	O
of	O
the	O
common	O
discretization	B-Algorithm
techniques	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
finite	B-Application
element	I-Application
method	I-Application
may	O
be	O
recast	O
as	O
a	O
multigrid	O
method	O
.	O
</s>
<s>
The	O
finite	B-Algorithm
difference	I-Algorithm
method	I-Algorithm
is	O
often	O
regarded	O
as	O
the	O
simplest	O
method	O
to	O
learn	O
and	O
use	O
.	O
</s>
<s>
The	O
finite	B-Application
element	I-Application
and	O
finite	O
volume	O
methods	O
are	O
widely	O
used	O
in	O
engineering	O
and	O
in	O
computational	O
fluid	O
dynamics	O
,	O
and	O
are	O
well	O
suited	O
to	O
problems	O
in	O
complicated	O
geometries	O
.	O
</s>
