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We now study the linear systems resulting from the
hp
-dG method (
12.10
). We
show that they may be solved iteratively, without causing a loss in the rates of
convergence in the error estimate (
12.11
), by the use of an incomplete GMRES
method. Furthermore, we prove that the overall complexity is linear (up to logarith-
mic terms).
12.3.1 Derivation of the Linear Systems
The
hp
-dG time stepping scheme (
12.10
) corresponds to a linear system of size
(r
m
+
1
)N
L
to be solved in each time step
m
=
1
,...,M
.Let
{
ϕ
j
:
j
=
0
,...,r
m
}
be
a basis of the polynomial space
S
r
m
(
−
1
,
1
)
. We also refer to
ϕ
j
as the reference time
shape functions. On the time interval
J
m
=
(t
m
−
1
,t
m
)
, the time shape functions
ϕ
m,j
are then defined as
ϕ
m,j
=
ϕ
j
◦
F
−
1
, where
F
m
is the mapping from the reference
m
interval
(
−
1
,
1
)
to
J
m
given by
1
2
(t
m
−
1
+
1
2
k
m
t.
F
m
(
t)
=
t
m
)
+
Since the semi-discrete approximation
U
|
J
m
in (
12.10
)
are both in
S
r
m
(J
m
,V
L
)
, they can be written in terms of the basis
|
J
m
and the test function
W
{
ϕ
m,j
:
j
=
0
,...,r
m
}
,
r
m
r
m
U
|
J
m
(x, t)
=
U
m,j
(x)ϕ
m,j
(t),
W
|
J
m
(x, t)
=
W
m,j
(x)ϕ
m,j
(t).
j
=
0
j
=
0
We choose normalized Legendre polynomials as reference time shape functions, i.e.
j
ϕ
j
(
t)
=
+
·
L
j
(
t),
∈ N
0
,
1
/
2
j
(12.12)
where
L
j
are the usual Legendre polynomials of degree
j
on
(
−
1
,
1
)
.
Example 12.3.3
The first four reference time shape functions of the form (
12.12
)
are
1
/
2
,
ϕ
0
(
t)
=
3
/
2
ϕ
1
(
t)
·
t,
=
5
/
2
·
(
3
t
2
ϕ
2
(
t)
=
−
1
)/
2
,
7
/
2
(
5
t
3
ϕ
3
(
t)
=
·
−
3
t)/
2
.
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