<s>
by	O
introducing	O
additional	O
stabilization	O
terms	O
in	O
the	O
Navier	O
–	O
Stokes	O
Galerkin	B-Algorithm
formulation	I-Algorithm
.	O
</s>
<s>
The	O
finite	B-Application
element	I-Application
(	O
FE	O
)	O
numerical	O
computation	O
of	O
incompressible	O
Navier	O
–	O
Stokes	O
equations	O
(	O
NS	O
)	O
suffers	O
from	O
two	O
main	O
sources	O
of	O
numerical	B-Algorithm
instabilities	I-Algorithm
arising	O
from	O
the	O
associated	O
Galerkin	O
problem	O
.	O
</s>
<s>
Equal	O
order	O
finite	B-Application
elements	I-Application
for	O
pressure	O
and	O
velocity	O
,	O
(	O
for	O
example	O
,	O
)	O
,	O
do	O
not	O
satisfy	O
the	O
inf-sup	O
condition	O
and	O
leads	O
to	O
instability	O
on	O
the	O
discrete	O
pressure	O
(	O
also	O
called	O
spurious	O
pressure	O
)	O
.	O
</s>
<s>
Moreover	O
,	O
the	O
advection	O
term	O
in	O
the	O
Navier	O
–	O
Stokes	O
equations	O
can	O
produce	O
oscillations	B-Algorithm
in	O
the	O
velocity	O
field	O
(	O
also	O
called	O
spurious	O
velocity	O
)	O
.	O
</s>
<s>
Such	O
spurious	O
velocity	O
oscillations	B-Algorithm
become	O
more	O
evident	O
for	O
advection-dominated	O
(	O
i.e.	O
,	O
high	O
Reynolds	O
number	O
)	O
flows	O
.	O
</s>
<s>
To	O
control	O
instabilities	B-Algorithm
arising	O
from	O
inf-sup	O
condition	O
and	O
convection	O
dominated	O
problem	O
,	O
pressure-stabilizing	O
Petrov	O
–	O
Galerkin	O
(	O
PSPG	O
)	O
stabilization	O
along	O
with	O
Streamline-Upwind	O
Petrov-Galerkin	O
(	O
SUPG	O
)	O
stabilization	O
can	O
be	O
added	O
to	O
the	O
NS	O
Galerkin	B-Algorithm
formulation	I-Algorithm
.	O
</s>
<s>
where	O
is	O
the	O
outward	O
directed	O
unit	O
normal	O
vector	O
to	O
,	O
is	O
the	O
Cauchy	O
stress	O
tensor	B-Device
,	O
is	O
the	O
fluid	O
density	O
,	O
and	O
and	O
are	O
the	O
usual	O
gradient	O
and	O
divergence	B-Application
operators	O
.	O
</s>
<s>
For	O
a	O
Newtonian	O
fluid	O
,	O
the	O
Cauchy	O
stress	O
tensor	B-Device
depends	O
linearly	O
on	O
the	O
components	O
of	O
the	O
strain	O
rate	O
tensor	B-Device
:	O
</s>
<s>
The	O
first	O
of	O
the	O
NS	O
equations	O
represents	O
the	O
balance	B-Algorithm
of	I-Algorithm
the	I-Algorithm
momentum	I-Algorithm
and	O
the	O
second	O
one	O
the	O
conservation	O
of	O
the	O
mass	O
,	O
also	O
called	O
continuity	O
equation	O
(	O
or	O
incompressible	O
constraint	O
)	O
.	O
</s>
<s>
The	O
weak	B-Algorithm
formulation	I-Algorithm
of	O
the	O
strong	O
formulation	O
of	O
the	O
NS	O
equations	O
is	O
obtained	O
by	O
multiplying	O
the	O
first	O
two	O
NS	O
equations	O
by	O
test	O
functions	O
and	O
,	O
respectively	O
,	O
belonging	O
to	O
suitable	O
function	B-Algorithm
spaces	I-Algorithm
,	O
and	O
integrating	O
these	O
equation	O
all	O
over	O
the	O
fluid	O
domain	O
.	O
</s>
<s>
Regarding	O
the	O
choice	O
of	O
the	O
function	B-Algorithm
spaces	I-Algorithm
,	O
it	O
's	O
enough	O
that	O
and	O
,	O
and	O
,	O
and	O
their	O
derivative	B-Algorithm
,	O
and	O
are	O
square-integrable	B-Algorithm
functionss	O
in	O
order	O
to	O
have	O
sense	O
in	O
the	O
integrals	O
that	O
appear	O
in	O
the	O
above	O
formulation	O
.	O
</s>
<s>
The	O
weak	B-Algorithm
formulation	I-Algorithm
of	O
Navier	O
–	O
Stokes	O
equations	O
reads	O
:	O
</s>
<s>
In	O
order	O
to	O
numerically	O
solve	O
the	O
NS	O
problem	O
,	O
first	O
the	O
discretization	B-Algorithm
of	O
the	O
weak	B-Algorithm
formulation	I-Algorithm
is	O
performed	O
.	O
</s>
<s>
Consider	O
a	O
triangulation	B-Algorithm
,	O
composed	O
by	O
tetrahedra	O
,	O
with	O
(	O
where	O
is	O
the	O
total	O
number	O
of	O
tetrahedra	O
)	O
,	O
of	O
the	O
domain	O
and	O
is	O
the	O
characteristic	O
length	O
of	O
the	O
element	O
of	O
the	O
triangulation	B-Algorithm
.	O
</s>
<s>
Introducing	O
two	O
families	O
of	O
finite-dimensional	O
sub-spaces	O
and	O
,	O
approximations	O
of	O
and	O
respectively	O
,	O
and	O
depending	O
on	O
a	O
discretization	B-Algorithm
parameter	O
,	O
with	O
and	O
,	O
</s>
<s>
the	O
discretized-in-space	O
Galerkin	O
problem	O
of	O
the	O
weak	O
NS	O
equation	O
reads	O
:	O
</s>
<s>
Time	O
discretization	B-Algorithm
of	O
discretized-in-space	O
NS	O
Galerkin	O
problem	O
can	O
be	O
performed	O
,	O
for	O
example	O
,	O
by	O
using	O
the	O
second	O
order	O
Backward	B-Algorithm
Differentiation	I-Algorithm
Formula	I-Algorithm
(	O
BDF2	O
)	O
,	O
that	O
is	O
an	O
implicit	B-Algorithm
second	O
order	O
multistep	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Thus	O
,	O
the	O
BDF2	O
approximation	O
of	O
the	O
time	O
derivative	B-Algorithm
reads	O
as	O
follows	O
:	O
</s>
<s>
So	O
,	O
the	O
fully	O
discretized	B-Algorithm
in	O
time	O
and	O
space	O
NS	O
Galerkin	O
problem	O
is	O
:	O
</s>
<s>
The	O
main	O
issue	O
of	O
a	O
fully	O
implicit	B-Algorithm
method	I-Algorithm
for	O
the	O
NS	O
Galerkin	B-Algorithm
formulation	I-Algorithm
is	O
that	O
the	O
resulting	O
problem	O
is	O
still	O
non	O
linear	O
,	O
due	O
to	O
the	O
convective	O
term	O
,	O
.	O
</s>
<s>
In	O
order	O
to	O
reduce	O
this	O
cost	O
,	O
it	O
is	O
possible	O
to	O
use	O
a	O
semi-implicit	B-Algorithm
approach	O
with	O
a	O
second	O
order	O
extrapolation	O
for	O
the	O
velocity	O
,	O
,	O
in	O
the	O
convective	O
term	O
:	O
</s>
<s>
with	O
,	O
and	O
independent	O
of	O
the	O
mesh	B-Algorithm
size	O
This	O
property	O
is	O
necessary	O
for	O
the	O
well	B-Algorithm
posedness	I-Algorithm
of	O
the	O
discrete	O
problem	O
and	O
the	O
optimal	B-Algorithm
convergence	I-Algorithm
of	I-Algorithm
the	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
equal	O
order	O
finite	B-Application
elements	I-Application
,	O
(	O
)	O
,	O
do	O
not	O
satisfy	O
the	O
inf-sup	O
condition	O
and	O
leads	O
to	O
instability	O
on	O
the	O
discrete	O
pressure	O
(	O
also	O
called	O
spurious	O
pressure	O
)	O
.	O
</s>
<s>
of	O
the	O
finite	B-Application
element	I-Application
for	O
the	O
velocity	O
and	O
pressure	O
field	O
,	O
respectively	O
.	O
</s>
<s>
Since	O
the	O
fully	O
discretized	B-Algorithm
Galerkin	O
problem	O
holds	O
for	O
all	O
elements	O
of	O
the	O
space	O
and	O
,	O
then	O
it	O
is	O
valid	O
also	O
for	O
the	O
basis	O
.	O
</s>
<s>
Hence	O
,	O
choosing	O
these	O
basis	O
functions	O
as	O
test	O
functions	O
in	O
the	O
fully	O
discretized	B-Algorithm
NS	O
Galerkin	O
problem	O
,	O
and	O
using	O
bilinearity	O
of	O
and	O
,	O
and	O
trilinearity	O
of	O
,	O
the	O
following	O
linear	O
system	O
is	O
obtained	O
:	O
</s>
<s>
NS	O
equations	O
with	O
finite	B-Application
element	I-Application
formulation	O
suffer	O
from	O
two	O
source	O
of	O
numerical	B-Algorithm
instability	I-Algorithm
,	O
due	O
to	O
the	O
fact	O
that	O
:	O
</s>
<s>
NS	O
is	O
a	O
convection	O
dominated	O
problem	O
,	O
which	O
means	O
"	O
large	O
"	O
,	O
where	O
numerical	O
oscillations	B-Algorithm
in	O
the	O
velocity	O
field	O
can	O
occur	O
(	O
spurious	O
velocity	O
)	O
;	O
</s>
<s>
FE	O
spaces	O
are	O
unstable	O
combinations	O
of	O
velocity	O
and	O
pressure	O
finite	B-Application
element	I-Application
spaces	O
,	O
that	O
do	O
not	O
satisfy	O
the	O
inf-sup	O
condition	O
,	O
and	O
generates	O
numerical	O
oscillations	B-Algorithm
in	O
the	O
pressure	O
field	O
(	O
spurious	O
pressure	O
)	O
.	O
</s>
<s>
To	O
control	O
instabilities	B-Algorithm
arising	O
from	O
inf-sup	O
condition	O
and	O
convection	O
dominated	O
problem	O
,	O
Pressure-Stabilizing	O
Petrov	O
–	O
Galerkin(PSPG )	O
stabilization	O
along	O
with	O
Streamline-Upwind	O
Petrov	O
–	O
Galerkin	O
(	O
SUPG	O
)	O
stabilization	O
can	O
be	O
added	O
to	O
the	O
NS	O
Galerkin	B-Algorithm
formulation	I-Algorithm
.	O
</s>
<s>
where	O
is	O
a	O
positive	O
constant	O
,	O
is	O
a	O
stabilization	O
parameter	O
,	O
is	O
a	O
generic	O
tetrahedron	O
belonging	O
to	O
the	O
finite	B-Application
elements	I-Application
partitioned	O
domain	O
,	O
is	O
the	O
residual	O
of	O
the	O
NS	O
equations	O
.	O
</s>
<s>
Since	O
it	O
is	O
based	O
on	O
the	O
residual	O
of	O
the	O
NS	O
equations	O
,	O
the	O
SUPG-PSPG	O
is	O
a	O
strongly	O
consistent	B-Algorithm
stabilization	O
method	O
.	O
</s>
<s>
The	O
discretized	B-Algorithm
finite	B-Application
element	I-Application
Galerkin	B-Algorithm
formulation	I-Algorithm
with	O
SUPG-PSPG	O
stabilization	O
can	O
be	O
written	O
as	O
:	O
</s>
<s>
and	O
,	O
and	O
are	O
two	O
stabilization	O
parameters	O
for	O
the	O
momentum	B-Algorithm
and	O
the	O
continuity	O
NS	O
equations	O
,	O
respectively	O
.	O
</s>
<s>
In	O
addition	O
,	O
the	O
notation	O
has	O
been	O
introduced	O
,	O
and	O
was	O
defined	O
in	O
agreement	O
with	O
the	O
semi-implicit	B-Algorithm
treatment	O
of	O
the	O
convective	O
term	O
.	O
</s>
<s>
The	O
other	O
terms	O
occur	O
to	O
obtain	O
a	O
strongly	O
consistent	B-Algorithm
stabilization	O
.	O
</s>
<s>
where	O
:	O
is	O
a	O
constant	O
obtained	O
by	O
an	O
inverse	O
inequality	O
relation	O
(	O
and	O
is	O
the	O
order	O
of	O
the	O
chosen	O
pair	O
)	O
;	O
is	O
a	O
constant	O
equal	O
to	O
the	O
order	O
of	O
the	O
time	O
discretization	B-Algorithm
;	O
is	O
the	O
time	O
step	O
;	O
is	O
the	O
"	O
element	O
length	O
"	O
(	O
e.g.	O
</s>
