<s>
In	O
applied	O
mathematics	O
,	O
discontinuous	B-Algorithm
Galerkin	I-Algorithm
methods	I-Algorithm
(	O
DG	B-Algorithm
methods	I-Algorithm
)	O
form	O
a	O
class	O
of	O
numerical	B-General_Concept
methods	I-General_Concept
for	O
solving	O
differential	O
equations	O
.	O
</s>
<s>
They	O
combine	O
features	O
of	O
the	O
finite	B-Application
element	I-Application
and	O
the	O
finite	B-Algorithm
volume	I-Algorithm
framework	O
and	O
have	O
been	O
successfully	O
applied	O
to	O
hyperbolic	O
,	O
elliptic	O
,	O
parabolic	O
and	O
mixed	O
form	O
problems	O
arising	O
from	O
a	O
wide	O
range	O
of	O
applications	O
.	O
</s>
<s>
DG	B-Algorithm
methods	I-Algorithm
have	O
in	O
particular	O
received	O
considerable	O
interest	O
for	O
problems	O
with	O
a	O
dominant	O
first-order	O
part	O
,	O
e.g.	O
</s>
<s>
Discontinuous	B-Algorithm
Galerkin	I-Algorithm
methods	I-Algorithm
were	O
first	O
proposed	O
and	O
analyzed	O
in	O
the	O
early	O
1970s	O
as	O
a	O
technique	O
to	O
numerically	B-General_Concept
solve	O
partial	O
differential	O
equations	O
.	O
</s>
<s>
In	O
1973	O
Reed	O
and	O
Hill	O
introduced	O
a	O
DG	B-Algorithm
method	I-Algorithm
to	O
solve	O
the	O
hyperbolic	O
neutron	O
transport	O
equation	O
.	O
</s>
<s>
The	O
origin	O
of	O
the	O
DG	B-Algorithm
method	I-Algorithm
for	O
elliptic	O
problems	O
cannot	O
be	O
traced	O
back	O
to	O
a	O
single	O
publication	O
as	O
features	O
such	O
as	O
jump	O
penalization	O
in	O
the	O
modern	O
sense	O
were	O
developed	O
gradually	O
.	O
</s>
<s>
DG	B-Algorithm
methods	I-Algorithm
for	O
elliptic	O
problems	O
were	O
already	O
developed	O
in	O
a	O
paper	O
by	O
Garth	O
Baker	O
in	O
the	O
setting	O
of	O
4th	O
order	O
equations	O
in	O
1977	O
.	O
</s>
<s>
A	O
more	O
complete	O
account	O
of	O
the	O
historical	O
development	O
and	O
an	O
introduction	O
to	O
DG	B-Algorithm
methods	I-Algorithm
for	O
elliptic	O
problems	O
is	O
given	O
in	O
a	O
publication	O
by	O
Arnold	O
,	O
Brezzi	O
,	O
Cockburn	O
and	O
Marini	O
.	O
</s>
<s>
A	O
number	O
of	O
research	O
directions	O
and	O
challenges	O
on	O
DG	B-Algorithm
methods	I-Algorithm
are	O
collected	O
in	O
the	O
proceedings	O
volume	O
edited	O
by	O
Cockburn	O
,	O
Karniadakis	O
and	O
Shu	O
.	O
</s>
<s>
Much	O
like	O
the	O
continuous	O
Galerkin	O
(	O
CG	O
)	O
method	O
,	O
the	O
discontinuous	O
Galerkin	O
(	O
DG	O
)	O
method	O
is	O
a	O
finite	B-Application
element	I-Application
method	I-Application
formulated	O
relative	O
to	O
a	O
weak	B-Algorithm
formulation	I-Algorithm
of	O
a	O
particular	O
model	O
system	O
.	O
</s>
<s>
Unlike	O
traditional	O
CG	O
methods	O
that	O
are	O
conforming	B-Application
,	O
the	O
DG	B-Algorithm
method	I-Algorithm
works	O
over	O
a	O
trial	O
space	B-Algorithm
of	I-Algorithm
functions	I-Algorithm
that	O
are	O
only	O
piecewise	B-Algorithm
continuous	I-Algorithm
,	O
and	O
thus	O
often	O
comprise	O
more	O
inclusive	O
function	B-Algorithm
spaces	I-Algorithm
than	O
the	O
finite-dimensional	O
inner	O
product	O
subspaces	O
utilized	O
in	O
conforming	B-Application
methods	O
.	O
</s>
<s>
We	O
derive	O
the	O
basis	O
representation	O
for	O
the	O
function	B-Algorithm
space	I-Algorithm
of	O
our	O
solution	O
.	O
</s>
<s>
The	O
term	O
is	O
chosen	O
to	O
be	O
equal	O
to	O
for	O
the	O
symmetric	O
interior	O
penalty	O
Galerkin	B-Algorithm
method	I-Algorithm
;	O
it	O
is	O
equal	O
to	O
for	O
the	O
non-symmetric	O
interior	O
penalty	O
Galerkin	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
direct	O
discontinuous	O
Galerkin	O
(	O
DDG	O
)	O
method	O
is	O
a	O
new	O
discontinuous	B-Algorithm
Galerkin	I-Algorithm
method	I-Algorithm
for	O
solving	O
diffusion	O
problems	O
.	O
</s>
<s>
The	O
advantages	O
of	O
this	O
method	O
compared	O
with	O
Discontinuous	B-Algorithm
Galerkin	I-Algorithm
method	I-Algorithm
is	O
that	O
the	O
direct	O
discontinuous	B-Algorithm
Galerkin	I-Algorithm
method	I-Algorithm
derives	O
the	O
numerical	B-General_Concept
format	O
by	O
directly	O
taking	O
the	O
numerical	B-General_Concept
flux	O
of	O
the	O
function	O
and	O
the	O
first	O
derivative	O
term	O
without	O
introducing	O
intermediate	O
variables	O
.	O
</s>
<s>
We	O
still	O
can	O
get	O
a	O
reasonable	O
numerical	B-General_Concept
results	O
by	O
using	O
this	O
method	O
,	O
and	O
the	O
derivation	O
process	O
is	O
more	O
simple	O
,	O
the	O
amount	O
of	O
calculation	O
is	O
greatly	O
reduced	O
.	O
</s>
<s>
The	O
direct	O
discontinuous	O
finite	B-Application
element	I-Application
method	I-Application
is	O
a	O
branch	O
of	O
the	O
Discontinuous	B-Algorithm
Galerkin	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
It	O
mainly	O
includes	O
transforming	O
the	O
problem	O
into	O
variational	O
form	O
,	O
regional	O
unit	O
splitting	O
,	O
constructing	O
basis	O
functions	O
,	O
forming	O
and	O
solving	O
discontinuous	O
finite	B-Application
element	I-Application
equations	O
,	O
and	O
convergence	O
and	O
error	O
analysis	O
.	O
</s>
<s>
Here	O
is	O
arbitrary	O
,	O
the	O
exact	O
solution	O
of	O
the	O
equation	O
is	O
replaced	O
by	O
the	O
approximate	O
solution	O
,	O
that	O
is	O
to	O
say	O
,	O
the	O
numerical	B-General_Concept
solution	I-General_Concept
we	O
need	O
is	O
obtained	O
by	O
solving	O
the	O
differential	O
equations	O
.	O
</s>
<s>
Choosing	O
a	O
proper	O
numerical	B-General_Concept
flux	O
is	O
critical	O
for	O
the	O
accuracy	O
of	O
DDG	O
method	O
.	O
</s>
<s>
The	O
numerical	B-General_Concept
flux	O
needs	O
to	O
satisfy	O
the	O
following	O
conditions	O
:	O
</s>
<s>
♦	O
The	O
numerical	B-General_Concept
flux	O
is	O
conservative	O
in	O
the	O
single	O
value	O
on	O
.	O
</s>
<s>
Thus	O
,	O
a	O
general	O
scheme	O
for	O
numerical	B-General_Concept
flux	O
is	O
given	O
:	O
</s>
<s>
Denote	O
that	O
the	O
error	O
between	O
the	O
exact	O
solution	O
and	O
the	O
numerical	B-General_Concept
solution	I-General_Concept
is	O
.	O
</s>
