<s>
In	O
numerical	B-General_Concept
mathematics	I-General_Concept
,	O
the	B-Algorithm
gradient	I-Algorithm
discretisation	I-Algorithm
method	I-Algorithm
(	O
GDM	O
)	O
is	O
a	O
framework	O
which	O
contains	O
classical	O
and	O
recent	O
numerical	O
schemes	O
for	O
diffusion	O
problems	O
of	O
various	O
kinds	O
:	O
linear	O
or	O
non-linear	O
,	O
steady-state	O
or	O
time-dependent	O
.	O
</s>
<s>
The	O
GDM	O
is	O
then	O
in	O
this	O
case	O
a	O
nonconforming	O
method	O
for	O
the	O
approximation	O
of	O
(	O
2	O
)	O
,	O
which	O
includes	O
the	O
nonconforming	O
finite	B-Application
element	I-Application
method	I-Application
.	O
</s>
<s>
The	O
"	O
mass-lumped	O
"	O
finite	B-Application
element	I-Application
case	O
enters	O
the	O
framework	O
of	O
the	O
GDM	O
,	O
replacing	O
by	O
,	O
where	O
is	O
a	O
dual	O
cell	O
centred	O
on	O
the	O
vertex	O
indexed	O
by	O
.	O
</s>
<s>
On	O
a	O
mesh	O
which	O
is	O
a	O
conforming	O
set	O
of	O
simplices	O
of	O
,	O
the	O
nonconforming	O
finite	B-Application
elements	I-Application
are	O
defined	O
by	O
the	O
basis	O
of	O
the	O
functions	O
which	O
are	O
affine	O
in	O
any	O
,	O
and	O
whose	O
value	O
at	O
the	O
centre	O
of	O
gravity	O
of	O
one	O
given	O
face	O
of	O
the	O
mesh	O
is	O
1	O
and	O
0	O
at	O
all	O
the	O
others	O
(	O
these	O
finite	B-Application
elements	I-Application
are	O
used	O
in	O
[	O
Crouzeix	O
et	O
al ]	O
for	O
the	O
approximation	O
of	O
the	O
Stokes	O
and	O
Navier-Stokes	O
equations	O
)	O
.	O
</s>
<s>
Then	O
the	O
method	O
enters	O
the	O
GDM	O
framework	O
with	O
the	O
same	O
definition	O
as	O
in	O
the	O
case	O
of	O
the	O
Galerkin	B-Algorithm
method	I-Algorithm
,	O
except	O
for	O
the	O
fact	O
that	O
must	O
be	O
understood	O
as	O
the	O
"	O
broken	O
gradient	O
"	O
of	O
,	O
in	O
the	O
sense	O
that	O
it	O
is	O
the	O
piecewise	O
constant	O
function	O
equal	O
in	O
each	O
simplex	O
to	O
the	O
gradient	O
of	O
the	O
affine	O
function	O
in	O
the	O
simplex	O
.	O
</s>
<s>
The	O
mixed	B-Algorithm
finite	I-Algorithm
element	I-Algorithm
method	I-Algorithm
consists	O
in	O
defining	O
two	O
discrete	O
spaces	O
,	O
one	O
for	O
the	O
approximation	O
of	O
and	O
another	O
one	O
for	O
.	O
</s>
<s>
Using	O
the	O
low	O
degree	O
Raviart	B-Algorithm
–	I-Algorithm
Thomas	I-Algorithm
basis	I-Algorithm
functions	I-Algorithm
allows	O
to	O
get	O
the	O
piecewise	O
constant	O
reconstruction	O
property	O
.	O
</s>
<s>
The	O
Discontinuous	O
Galerkin	B-Algorithm
method	I-Algorithm
consists	O
in	O
approximating	O
problems	O
by	O
a	O
piecewise	O
polynomial	O
function	O
,	O
without	O
requirements	O
on	O
the	O
jumps	O
from	O
an	O
element	O
to	O
the	O
other	O
.	O
</s>
