### 3.2. The Monotone Iterative Method

Assume that

from (3.2) satisfies the two-sided constraint

We consider the following iterative method for solving (3.2). Choose an initial mesh function

. On each time level, the iterative sequence

,

, is defined by the recurrence formulae

where
is the residual of the difference scheme (3.2) on
.

On a time level

, we say that

is an upper solution of (3.2) with respect to

if it satisfies the inequalities

Similarly,

is called a lower solution if it satisfies all the reversed inequalities. Upper and lower solutions satisfy the inequality

This result can be proved in a similar way as for the elliptic problem.

The following theorem gives the monotone property of the iterative method (3.7).

Theorem 3.2.

Assume that

satisfies (3.6). Let

be given and

,

be upper and lower solutions of (3.2) corresponding

. Then the upper sequence

generated by (3.7) converges monotonically from above to the unique solution

of the problem

the lower sequence

generated by (3.7) converges monotonically from below to

and the following inequalities hold

Proof.

We consider only the case of the upper sequence, and the case of the lower sequence can be proved in a similar way.

If
is an upper solution, then from (3.7) we conclude that

From Lemma 3.1, it follows that

and from (3.7), it follows that
satisfies the boundary conditions.

Using the mean-value theorem and the equation for
from (3.7), we represent
in the form

where
,
. Since the mesh function
is nonpositive on
and taking into account (3.6), we conclude that
is an upper solution to (3.2). By induction on
, we obtain that
,
,
, and prove that
is a monotonically decreasing sequence of upper solutions.

We now prove that the monotone sequence
converges to the solution of (3.2). The sequence
is monotonically decreasing and bounded below by
, where
is any lower solution (3.9). Now by linearity of the operator
and the continuity of
, we have also from (3.7) that the mesh function
defined by

is an exact solution to (3.2). If by contradiction, we assume that there exist two solutions

and

to (3.2), then by the mean-value theorem, the difference

satisfies the system

By Lemma 3.1,
which leads to the uniqueness of the solution to (3.2). This proves the theorem.

Consider the following approach for constructing initial upper and lower solutions

and

. Introduce the difference problems

The functions
,
are upper and lower solutions, respectively. This result can be proved in a similar way as for the elliptic problem.

Theorem 3.3.

Let initial upper or lower solution be chosen in the form of (3.17), and let

satisfy (3.6). Suppose that on each time level the number of iterates

. Then for the monotone iterative methods (3.7), the following estimate on convergence rate holds:

where
is the solution to (3.2),
, and constant
is independent of
,
and
.

Proof.

Similar to (3.14), using the mean-value theorem and the equation for

from (3.7), we have

From here and (3.7), we have

Using (3.5) and (3.6), we have

where
is defined in (3.18).

Introduce the notation

where

. Using the mean-value theorem, from (3.2) and (3.19), we conclude that

satisfies the problem

where

, and we have taken into account that

. By (3.5), (3.6), and (3.21),

Using (3.6), (3.17), and the mean-value theorem, estimate
from (3.7) by (3.5),

where

is independent of

(

),

and

. Thus,

Similarly, from (3.2) and (3.19), it follows that

Using (3.17), estimate

from (3.7) by (3.5),

where

. As follows from Theorem 3.2, the monotone sequences

and

are bounded from above and below by, respectively,

and

. Applying (3.5) to problem (3.17) at

, we have

where constant

is independent of

,

and

. Thus, we prove that

is independent of

,

and

. From (3.26) and (3.28), we conclude

By induction on

, we prove

where all constants
are independent of
,
and
. Taking into account that
, we prove the estimate (3.18) with
.

In [

4], we prove that the difference scheme (3.2) on the piecewise uniform mesh (2.8) converges

-uniformly to the solution of problem (1.2):

where
is the exact solution to (3.2), and constant
is independent of
,
and
. From here and Theorem 3.3, we conclude the following theorem.

Theorem 3.4.

Suppose that on each time level the initial upper or lower solution

is chosen in the form of (3.17) and

. Then the monotone iterative method (3.7) on the piecewise uniform mesh (2.8) converges

-uniformly to the solution of problem (1.2):

where
, and constant
is independent of
,
and
.