In this section we construct fully discrete discontinuous Galerkin approximations using extrapolated Crank-Nicolson method and prove the optimal convergence in
normed space.
For a positive integer
we let
and for
and we define
and
. For
, we define
,
and
.
The extrapolated Crank-Nicolson discontinuous Galerkin approximation
is defined by
where
,
.
To apply (4.1), we need two initial stages
and
to be defined in the following:
where
.
To prove the optimal convergence of
in
normed space we denote
and
,
,
,
,
.
Now we state the following approximations for
whose proofs can be found in [18, 19].
Theorem 4.1.
If
and
then there exists a constant
independent of
and
satisfying
(i)
(ii)
.
Theorem 4.2.
If
,
,
and
then there exists a constant
independent of
and
satisfying
(i)
(ii)
provided that
.
By simple computations and the applications of Theorem 4.2 we obtain the following lemmas.
Lemma 4.3.
If
satisfies
then there exists a constant
independent of
and
such that
Consequently from Lemma 4.3 there exists a constant
independent of
and
such that
if
is sufficiently smooth.
Lemma 4.4.
If
then there exists a constant
independent of
and
such that
Consequently from Lemma 4.4 there exists a constant
independent of
and
such that
if
is sufficiently smooth.
Lemma 4.5.
If we let
then there exists a constant
independent of
and
such that
Consequently from Lemma 4.5, we induce that there exists a constant
independent of
and
such that
if
is sufficiently smooth.
Theorem 4.6.
For
and
, if
,
then there exists a constant
independent of
and
such that for
,
,
, 
hold where
and
.
Proof.
From (4.1) and (1.1), we have
By the notations of
and
, we get
By the definition of
, we obtain
From the definition of
, we have
Substituting (4.12)–(4.14) in (4.11) and choosing
imply that
By Cauchy-Schwarz'sinequality clearly we have
By the definition of
we have
For the definition of
we get
Applying (4.17) and (4.18) in (4.15) we conclude that
For sufficiently small
by applying Lemma 4.3 there exists a constant
such that
Applying Lemmas 4.3 and 4.4,
can be estimated as follows:
We obtain the following estimates of
for each 
From the definition of
, we can separate
as follows:
By applying Lemma 4.5,
can be estimated in the following way:
Similarly there exists a constant
such that
By applying the trace inequality we have
From the estimation of
,
, we have
By applying Lemma 4.3 we obviously obtain
Now we can separate
as follows:
Since
can be estimated as follows
We apply Lemma 3.2 to estimate
as follows:
From the result of approximation of
of Theorem 4.1
Therefore we get
Similarly,
and
are estimated as follows:
Substituting the estimates of
,
into (4.19), we get
If we sum both sides of (4.36) from
to
, then we obtain
which implies
where
is sufficiently small. By applying the discrete version of Gronwall's inequality, we have
Therefore by applying the result of Lemma 4.7 we have
which proves the optimal
error estimation of the fully discrete solutions.
Lemma 4.7 can be proved by the similar process of Theorem 4.6. as follows
Lemma 4.7.
For
and
, if
,
and
for some constant
then there exists a constant
independent of
and 