<s>
The	O
exponential	O
convergence	O
makes	O
the	O
method	O
very	O
attractive	O
compared	O
to	O
most	O
other	O
finite	B-Application
element	I-Application
methods	I-Application
,	O
which	O
only	O
converge	O
with	O
an	O
algebraic	O
rate	O
.	O
</s>
<s>
The	O
hp-FEM	B-Algorithm
differs	O
from	O
the	O
standard	O
(	O
lowest-order	O
)	O
FEM	O
in	O
many	O
aspects	O
.	O
</s>
<s>
Automatic	O
hp-adaptivity	O
:	O
In	O
hp-FEM	B-Algorithm
,	O
an	O
element	O
can	O
be	O
hp-refined	O
in	O
many	O
different	O
ways	O
,	O
such	O
as	O
:	O
Increasing	O
its	O
polynomial	O
degree	O
without	O
subdividing	O
it	O
in	O
space	O
,	O
or	O
subdividing	O
the	O
element	O
geometrically	O
,	O
where	O
various	O
polynomial	O
degrees	O
can	O
be	O
applied	O
to	O
the	O
sub-elements	O
.	O
</s>
<s>
By	O
contrast	O
,	O
the	O
stiffness	O
matrices	O
in	O
hp-FEM	B-Algorithm
typically	O
are	O
much	O
smaller	O
,	O
but	O
(	O
for	O
the	O
same	O
matrix	O
size	O
)	O
their	O
assembly	O
takes	O
more	O
time	O
than	O
in	O
standard	O
FEM	O
.	O
</s>
<s>
This	O
is	O
mainly	O
due	O
to	O
the	O
computational	O
cost	O
of	O
numerical	B-General_Concept
quadrature	O
,	O
which	O
must	O
have	O
higher	O
precision	O
,	O
and	O
therefore	O
be	O
of	O
higher	O
order	O
,	O
compared	O
to	O
standard	O
FEM	O
to	O
take	O
advantage	O
of	O
the	O
faster	O
convergence	O
rates	O
.	O
</s>
<s>
Analytical	O
challenges	O
:	O
hp-FEM	B-Algorithm
is	O
generally	O
considered	O
to	O
be	O
more	O
difficult	O
to	O
understand	O
from	O
the	O
analytical	O
point	O
of	O
view	O
than	O
standard	O
FEM	O
.	O
</s>
<s>
These	O
results	O
state	O
that	O
,	O
usually	O
with	O
some	O
limiting	O
assumptions	O
on	O
the	O
mesh	O
,	O
the	O
piecewise-polynomial	B-Algorithm
FEM	O
approximation	O
obeys	O
analogous	O
maximum	O
principles	O
as	O
the	O
underlying	O
elliptic	O
PDE	O
.	O
</s>
<s>
The	O
DMP	O
are	O
quite	O
well	O
understood	O
for	O
lowest-order	O
FEM	O
but	O
completely	O
unknown	O
for	O
the	O
hp-FEM	B-Algorithm
in	O
two	O
or	O
more	O
dimensions	O
.	O
</s>
<s>
Programming	O
challenges	O
:	O
It	O
is	O
much	O
harder	O
to	O
implement	O
an	O
hp-FEM	B-Algorithm
solver	O
than	O
standard	O
FEM	O
code	O
.	O
</s>
<s>
One	O
can	O
use	O
it	O
to	O
show	O
the	O
dramatic	O
difference	O
in	O
the	O
performance	O
of	O
standard	O
FEM	O
and	O
hp-FEM	B-Algorithm
.	O
</s>
<s>
The	O
knowledge	O
of	O
the	O
exact	O
solution	O
makes	O
it	O
possible	O
to	O
calculate	O
the	O
approximation	O
error	O
exactly	O
and	O
thus	O
compare	O
various	O
numerical	B-General_Concept
methods	I-General_Concept
.	O
</s>
<s>
For	O
illustration	O
,	O
the	O
problem	O
was	O
solved	O
using	O
three	O
different	O
versions	O
of	O
adaptive	O
FEM	O
:	O
with	O
linear	O
elements	O
,	O
quadratic	O
elements	O
,	O
and	O
hp-FEM	B-Algorithm
.	O
</s>
<s>
The	O
reader	O
can	O
see	O
in	O
the	O
graphs	O
that	O
the	O
convergence	O
of	O
the	O
hp-FEM	B-Algorithm
is	O
much	O
faster	O
than	O
the	O
convergence	O
of	O
both	O
other	O
methods	O
.	O
</s>
<s>
The	O
performance	O
gap	O
is	O
large	O
enough	O
that	O
the	O
linear	O
FEM	O
might	O
not	O
converge	O
at	O
all	O
(	O
in	O
reasonable	O
time	O
)	O
and	O
the	O
quadratic	O
FEM	O
would	O
need	O
hundreds	O
of	O
thousands	O
or	O
perhaps	O
millions	O
of	O
DOF	O
to	O
reach	O
the	O
accuracy	O
that	O
hp-FEM	B-Algorithm
attained	O
with	O
approximately	O
17,000	O
DOF	O
.	O
</s>
<s>
Obtaining	O
very	O
accurate	O
results	O
using	O
relatively	O
few	O
degrees	O
of	O
freedom	O
is	O
the	O
main	O
strength	O
of	O
hp-FEM	B-Algorithm
.	O
</s>
<s>
Smooth	O
functions	O
can	O
be	O
approximated	O
much	O
more	O
efficiently	O
using	O
large	O
high-order	O
elements	O
than	O
small	O
piecewise-linear	O
ones	O
.	O
</s>
<s>
This	O
means	O
that	O
the	O
quadratic	O
approximation	O
was	O
roughly	O
3.5-times	O
more	O
efficient	O
than	O
the	O
piecewise-linear	O
one	O
.	O
</s>
<s>
hp-FEM	B-Algorithm
is	O
based	O
on	O
an	O
optimal	O
combination	O
of	O
these	O
two	O
approaches	O
which	O
leads	O
to	O
exponential	O
convergence	O
.	O
</s>
<s>
As	O
soon	O
as	O
it	O
is	O
harder	O
to	O
program	O
and	O
parallelize	B-Operating_System
hp-FEM	B-Algorithm
compared	O
to	O
h-FEM	O
,	O
the	O
convergence	O
excellence	O
of	O
hp-refinement	O
may	O
become	O
impractical	O
.	O
</s>
<s>
Deal.II	B-Language
:	O
Deal.II	B-Language
is	O
a	O
free	O
,	O
open	O
source	O
library	O
to	O
solve	O
partial	O
differential	O
equations	O
using	O
the	O
finite	B-Application
element	I-Application
method	I-Application
.	O
</s>
<s>
:	O
C/C	O
++	O
hp-FEM/DGFEM/BEM	O
library	O
for	O
elliptic	O
equations	O
developed	O
at	O
SAM	O
,	O
ETH	O
Zurich	O
(	O
Switzerland	O
)	O
and	O
in	O
the	O
group	O
of	O
K	O
.	O
Schmidt	O
at	O
TU	O
Berlin	O
(	O
Germany	O
)	O
.	O
</s>
<s>
Finite	B-Application
element	I-Application
software	O
developed	O
at	O
the	O
National	O
Institute	O
for	O
Standards	O
and	O
Technology	O
,	O
USA	O
,	O
for	O
numerical	B-General_Concept
solution	I-General_Concept
of	O
2D	O
elliptic	O
partial	O
differential	O
equations	O
on	O
distributed	O
memory	O
parallel	B-Operating_System
computers	I-Operating_System
and	O
multicore	O
computers	O
using	O
adaptive	O
mesh	O
refinement	O
and	O
multigrid	O
solution	O
techniques	O
.	O
</s>
<s>
Hermes	B-Language
Project	I-Language
:	O
C/C	O
++	O
/Python	O
library	O
for	O
rapid	O
prototyping	O
of	O
space	O
-	O
and	O
space-time	O
adaptive	O
hp-FEM	B-Algorithm
solvers	O
for	O
a	O
large	O
variety	O
of	O
PDEs	O
and	O
multiphysics	O
PDE	O
systems	O
,	O
developed	O
by	O
the	O
hp-FEM	B-Algorithm
group	O
at	O
the	O
University	O
of	O
Nevada	O
,	O
Reno	O
(	O
USA	O
)	O
,	O
Institute	O
of	O
Thermo-mechanics	O
,	O
Prague	O
(	O
Czech	O
Republic	O
)	O
,	O
and	O
the	O
University	O
of	O
West	O
Bohemia	O
in	O
Pilsen	O
(	O
Czech	O
Republic	O
)	O
–	O
with	O
the	O
Agros2D	B-Language
engineering	O
software	O
built	O
on	O
top	O
of	O
the	O
Hermes	O
library	O
.	O
</s>
<s>
:	O
PHG	O
is	O
a	O
toolbox	O
for	O
developing	O
parallel	O
adaptive	O
finite	B-Application
element	I-Application
programs	O
.	O
</s>
<s>
It	O
's	O
suitable	O
for	O
h-	O
,	O
p	O
-	O
and	O
hp-fem	B-Algorithm
.	O
</s>
<s>
PHG	O
has	O
an	O
object	O
oriented	O
design	O
which	O
hides	O
parallelization	B-Operating_System
details	O
and	O
provides	O
common	O
operations	O
on	O
meshes	O
and	O
finite	B-Application
element	I-Application
functions	O
in	O
an	O
abstract	O
way	O
,	O
allowing	O
the	O
users	O
to	O
concentrate	O
on	O
their	O
numerical	B-General_Concept
algorithms	I-General_Concept
.	O
</s>
<s>
is	O
a	O
finite	B-Application
element	I-Application
analysis	I-Application
code	O
tailored	O
for	O
the	O
solution	O
of	O
multi-physics	O
problems	O
with	O
arbitrary	O
levels	O
of	O
approximation	O
,	O
different	O
levels	O
of	O
mesh	O
refinement	O
and	O
optimized	O
for	O
high-performance	O
computing	O
.	O
</s>
<s>
is	O
a	O
multi-physics	O
,	O
hp-adaptive	O
,	O
user-friendly	O
,	O
open	O
source	O
C++	O
finite	B-Application
element	I-Application
library	O
currently	O
developed	O
at	O
Tampere	O
University	O
,	O
Finland	O
.	O
</s>
<s>
It	O
combines	O
3D	O
tetrahedra	O
&	O
2D	O
triangle/quadrangle	O
conformal	O
adaptive	O
mesh	O
refinement	O
with	O
arbitrary	O
order	O
hierarchical	O
H1	O
&	O
H-curl	O
function	O
spaces	O
for	O
general	O
static	O
and	O
transient	O
hp-FEM	B-Algorithm
.	O
</s>
<s>
is	O
a	O
finite	B-Application
element	I-Application
analysis	I-Application
tool	O
with	O
hp-capabilities	O
oriented	O
towards	O
detailed	O
structural	O
analysis	O
.	O
</s>
