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