To investigate the convergence of the method, note that the error function , , is the solution of the discrete problem

where and are given by (4.10) and (4.13), respectively.

Lemma 5.1.

Let be approximate solution of (2.1). Then the following estimate holds

where

Proof.

The proof follows easily by induction in .

Lemma 5.2.

Let be solution of (5.1). Then following estimate holds

Proof.

It evidently follows from (5.2) by taking and .

Lemma 5.3.

Under the above assumptions of Section 2 and Lemma 3.1, for the error function , the following estimate holds

Proof.

To this end, it suffices to establish that the functions , involved in the expression for , admit the estimate

Using the mean value theorem, we get

Hence

and taking also into account that and using Lemma 3.1, we have

For , in view of and using Lemma 3.1, we obtain

Hence

and after replacing this reduces to

which yields

The same estimate is obtained for in the similar manner as above.

Combining the previous lemmas we get the following final estimate, that is, uniformly convergent estimate.

Theorem 5.4.

Let be the solution of (2.1) and be the solution of (4.14). Then the following estimate holds