<s>
In	O
the	O
numerical	O
solution	O
of	O
partial	O
differential	O
equations	O
,	O
a	O
topic	O
in	O
mathematics	O
,	O
the	O
spectral	B-Algorithm
element	I-Algorithm
method	I-Algorithm
(	O
SEM	O
)	O
is	O
a	O
formulation	O
of	O
the	O
finite	B-Application
element	I-Application
method	I-Application
(	O
FEM	O
)	O
that	O
uses	O
high	O
degree	O
piecewise	B-Algorithm
polynomials	I-Algorithm
as	O
basis	O
functions	O
.	O
</s>
<s>
The	O
spectral	B-Algorithm
element	I-Algorithm
method	I-Algorithm
was	O
introduced	O
in	O
a	O
1984	O
paper	O
by	O
A	O
.	O
T	O
.	O
Patera	O
.	O
</s>
<s>
The	O
spectral	B-Algorithm
method	I-Algorithm
expands	O
the	O
solution	O
in	O
trigonometric	O
series	O
,	O
a	O
chief	O
advantage	O
being	O
that	O
the	O
resulting	O
method	O
is	O
of	O
a	O
very	O
high	O
order	O
.	O
</s>
<s>
The	O
spectral	B-Algorithm
element	I-Algorithm
method	I-Algorithm
chooses	O
instead	O
a	O
high	O
degree	O
piecewise	B-Algorithm
polynomial	I-Algorithm
basis	O
functions	O
,	O
also	O
achieving	O
a	O
very	O
high	O
order	O
of	O
accuracy	O
.	O
</s>
<s>
Although	O
the	O
method	O
can	O
be	O
applied	O
with	O
a	O
modal	O
piecewise	B-Algorithm
orthogonal	O
polynomial	O
basis	O
,	O
it	O
is	O
most	O
often	O
implemented	O
with	O
a	O
nodal	O
tensor	O
product	O
Lagrange	O
basis	O
.	O
</s>
<s>
The	O
method	O
gains	O
its	O
efficiency	O
by	O
placing	O
the	O
nodal	O
points	O
at	O
the	O
Legendre-Gauss-Lobatto	O
(	O
LGL	O
)	O
points	O
and	O
performing	O
the	O
Galerkin	B-Algorithm
method	I-Algorithm
integrations	O
with	O
a	O
reduced	O
Gauss-Lobatto	B-Algorithm
quadrature	I-Algorithm
using	O
the	O
same	O
nodes	O
.	O
</s>
<s>
The	O
classic	O
analysis	O
of	O
Galerkin	B-Algorithm
methods	I-Algorithm
and	O
Céa	O
's	O
lemma	O
holds	O
here	O
and	O
it	O
can	O
be	O
shown	O
that	O
,	O
if	O
is	O
the	O
solution	O
of	O
the	O
weak	O
equation	O
,	O
is	O
the	O
approximate	O
solution	O
and	O
:	O
</s>
<s>
element	O
length	O
)	O
,	O
is	O
independent	O
from	O
,	O
and	O
is	O
no	O
larger	O
than	O
the	O
degree	O
of	O
the	O
piecewise	B-Algorithm
polynomial	I-Algorithm
basis	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
if	O
is	O
an	O
analytic	B-Language
function	I-Language
:	O
</s>
<s>
G-NI	O
or	O
SEM-NI	O
are	O
the	O
most	O
used	O
spectral	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
The	O
Galerkin	O
formulation	O
of	O
spectral	B-Algorithm
methods	I-Algorithm
or	O
spectral	B-Algorithm
element	I-Algorithm
methods	I-Algorithm
,	O
for	O
G-NI	O
or	O
SEM-NI	O
respectively	O
,	O
is	O
modified	O
and	O
Gauss-Lobatto	B-Algorithm
integration	I-Algorithm
is	O
used	O
instead	O
of	O
integrals	O
in	O
the	O
definition	O
of	O
the	O
bilinear	O
form	O
and	O
in	O
the	O
functional	O
.	O
</s>
<s>
SEM	O
is	O
a	O
Galerkin	O
based	O
FEM	O
(	O
finite	B-Application
element	I-Application
method	I-Application
)	O
with	O
Lagrange	O
basis	O
(	O
shape	O
)	O
functions	O
and	O
reduced	O
numerical	O
integration	O
by	O
Lobatto	B-Algorithm
quadrature	I-Algorithm
using	O
the	O
same	O
nodes	O
.	O
</s>
<s>
The	O
pseudospectral	B-Algorithm
method	I-Algorithm
,	O
orthogonal	O
collocation	O
,	O
differential	O
quadrature	O
method	O
,	O
and	O
G-NI	O
are	O
different	O
names	O
for	O
the	O
same	O
method	O
.	O
</s>
<s>
These	O
methods	O
employ	O
global	O
rather	O
than	O
piecewise	B-Algorithm
polynomial	I-Algorithm
basis	O
functions	O
.	O
</s>
<s>
The	O
extension	O
to	O
a	O
piecewise	B-Algorithm
FEM	O
or	O
SEM	O
basis	O
is	O
almost	O
trivial	O
.	O
</s>
<s>
The	O
spectral	B-Algorithm
element	I-Algorithm
method	I-Algorithm
uses	O
a	O
tensor	O
product	O
space	O
spanned	O
by	O
nodal	O
basis	O
functions	O
associated	O
with	O
Gauss	O
–	O
Lobatto	O
points	O
.	O
</s>
<s>
In	O
contrast	O
,	O
the	O
p-version	B-Algorithm
finite	I-Algorithm
element	I-Algorithm
method	I-Algorithm
spans	O
a	O
space	O
of	O
high	O
order	O
polynomials	O
by	O
nodeless	O
basis	O
functions	O
,	O
chosen	O
approximately	O
orthogonal	O
for	O
numerical	B-Algorithm
stability	I-Algorithm
.	O
</s>
<s>
Since	O
not	O
all	O
interior	O
basis	O
functions	O
need	O
to	O
be	O
present	O
,	O
the	O
p-version	B-Algorithm
finite	I-Algorithm
element	I-Algorithm
method	I-Algorithm
can	O
create	O
a	O
space	O
that	O
contains	O
all	O
polynomials	O
up	O
to	O
a	O
given	O
degree	O
with	O
fewer	O
degrees	O
of	O
freedom	O
.	O
</s>
<s>
However	O
,	O
some	O
speedup	O
techniques	O
possible	O
in	O
spectral	B-Algorithm
methods	I-Algorithm
due	O
to	O
their	O
tensor-product	O
character	O
are	O
no	O
longer	O
available	O
.	O
</s>
<s>
The	O
hp	O
finite	B-Application
element	I-Application
method	I-Application
(	O
hp-FEM	B-Algorithm
)	O
combines	O
the	O
advantages	O
of	O
the	O
h	O
and	O
p	O
refinements	O
to	O
obtain	O
exponential	O
convergence	O
rates	O
.	O
</s>
