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