Open Access

Robust Monotone Iterates for Nonlinear Singularly Perturbed Boundary Value Problems

Boundary Value Problems20092009:320606

DOI: 10.1155/2009/320606

Received: 8 April 2009

Accepted: 11 May 2009

Published: 14 May 2009

Abstract

This paper is concerned with solving nonlinear singularly perturbed boundary value problems. Robust monotone iterates for solving nonlinear difference scheme are constructed. Uniform convergence of the monotone methods is investigated, and convergence rates are estimated. Numerical experiments complement the theoretical results.

1. Introduction

We are interested in numerical solving of two nonlinear singularly perturbed problems of elliptic and parabolic types.

The first one is the elliptic problem
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq1_HTML.gif is a positive parameter, and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq2_HTML.gif is sufficiently smooth function. For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq3_HTML.gif this problem is singularly perturbed, and the solution has boundary layers near https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq4_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq5_HTML.gif (see [1] for details).

The second problem is the one-dimensional parabolic problem
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ2_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq6_HTML.gif is a positive parameter. Under suitable continuity and compatibility conditions on the data, a unique solution of this problem exists. For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq7_HTML.gif problem (1.2) is singularly perturbed and has boundary layers near the lateral boundary of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq8_HTML.gif (see [2] for details).

In the study of numerical methods for nonlinear singularly perturbed problems, the two major points to be developed are: (i) constructing robust difference schemes (this means that unlike classical schemes, the error does not increase to infinity, but rather remains bounded, as the small parameter approaches zero); (ii) obtaining reliable and efficient computing algorithms for solving nonlinear discrete problems.

Our goal is to construct a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq9_HTML.gif -uniform numerical method for solving problem (1.1), that is, a numerical method which generates https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq10_HTML.gif -uniformly convergent numerical approximations to the solution. We use a numerical method based on the classical difference scheme and the piecewise uniform mesh of Shishkin-type [3]. For solving problem (1.2), we use the implicit difference scheme based on the piecewise uniform mesh in the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq11_HTML.gif -direction, which converges https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq12_HTML.gif -uniformly [4].

A major point about the nonlinear difference schemes is to obtain reliable and efficient computational methods for computing the solution. The reliability of iterative techniques for solving nonlinear difference schemes can be essentially improved by using component-wise monotone globally convergent iterations. Such methods can be controlled every time. A fruitful method for the treatment of these nonlinear schemes is the method of upper and lower solutions and its associated monotone iterations [5]. Since an initial iteration in the monotone iterative method is either an upper or lower solution, which can be constructed directly from the difference equation without any knowledge of the exact solution, this method simplifies the search for the initial iteration as is often required in the Newton method. In the context of solving systems of nonlinear equations, the monotone iterative method belongs to the class of methods based on convergence under partial ordering (see [5, Chapter 13] for details).

The purpose of this paper is to construct https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq13_HTML.gif -uniformly convergent monotone iterative methods for solving https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq14_HTML.gif -uniformly convergent nonlinear difference schemes.

The structure of the paper is as follows. In Section 2, we prove that the classical difference scheme on the piecewise uniform mesh converges https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq15_HTML.gif -uniformly to the solution of problem (1.1). A robust monotone iterative method for solving the nonlinear difference scheme is constructed. In Section 3, we construct a robust monotone iterative method for solving problem (1.2). In the final Section 4, we present numerical experiments which complement the theoretical results.

2. The Elliptic Problem

The following lemma from [1] contains necessary estimates of the solution to (1.1).

Lemma 2.1.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq16_HTML.gif is the solution to (1.1), the following estimates hold:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ3_HTML.gif
(2.1)

where constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq17_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq18_HTML.gif .

For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq19_HTML.gif , the boundary layers appear near https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq20_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq21_HTML.gif .

2.1. The Nonlinear Difference Scheme

Introduce a nonuniform mesh https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq22_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ4_HTML.gif
(2.2)
For solving (1.1), we use the classical difference scheme
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ5_HTML.gif
(2.3)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq23_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq24_HTML.gif . We introduce the linear version of this problem
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ6_HTML.gif
(2.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq25_HTML.gif . We now formulate a discrete maximum principle for the difference operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq26_HTML.gif and give an estimate of the solution to (2.4).

Lemma 2.2.
  1. (i)
    If a mesh function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq27_HTML.gif satisfies the conditions
    https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ7_HTML.gif
    (2.5)
     
then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq28_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq29_HTML.gif .
  1. (ii)

    If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq30_HTML.gif , then the following estimate of the solution to (2.4) holds true:

     
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ8_HTML.gif
(2.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq31_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq32_HTML.gif .

The proof of the lemma can be found in [6].

2.2. Uniform Convergence on the Piecewise Uniform Mesh

We employ a layer-adapted mesh of a piecewise uniform type [3]. The piecewise uniform mesh is formed in the following manner. We divide the interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq33_HTML.gif into three parts https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq34_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq35_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq36_HTML.gif . Assuming that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq37_HTML.gif is divisible by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq38_HTML.gif , in the parts https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq39_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq40_HTML.gif we use the uniform mesh with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq41_HTML.gif mesh points, and in the part https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq42_HTML.gif the uniform mesh with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq43_HTML.gif mesh points is in use. The transition points https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq44_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq45_HTML.gif are determined by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ9_HTML.gif
(2.7)
This defines the piecewise uniform mesh. If the parameter https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq46_HTML.gif is small enough, then the uniform mesh inside of the boundary layers with the step size https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq47_HTML.gif is fine, and the uniform mesh outside of the boundary layers with the step size https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq48_HTML.gif is coarse, such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ10_HTML.gif
(2.8)

In the following theorem, we give the convergence property of the difference scheme (2.3).

Theorem 2.3.

The difference scheme (2.3) on the piecewise uniform mesh (2.8) converges https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq49_HTML.gif -uniformly to the solution of (1.1):
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ11_HTML.gif
(2.9)

where constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq50_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq51_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq52_HTML.gif .

Proof.

Using Green's function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq53_HTML.gif of the differential operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq54_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq55_HTML.gif , we represent the exact solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq56_HTML.gif in the form
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ12_HTML.gif
(2.10)
where the local Green function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq57_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ13_HTML.gif
(2.11)
and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq58_HTML.gif are defined by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ14_HTML.gif
(2.12)
Equating the derivatives https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq59_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq60_HTML.gif , we get the following integral-difference formula:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ15_HTML.gif
(2.13)
where here and below we suppress variable https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq61_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq62_HTML.gif . Representing https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq63_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq64_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq65_HTML.gif in the forms
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ16_HTML.gif
(2.14)
the above integral-difference formula can be written as
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ17_HTML.gif
(2.15)
where the truncation error https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq66_HTML.gif of the exact solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq67_HTML.gif to (1.1) is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ18_HTML.gif
(2.16)
From here, it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ19_HTML.gif
(2.17)

From Lemma 2.1, the following estimate on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq68_HTML.gif holds:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ20_HTML.gif
(2.18)
We estimate the truncation error https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq69_HTML.gif in (2.17) on the interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq70_HTML.gif . Consider the following three cases: https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq71_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq72_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq73_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq74_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq75_HTML.gif , and taking into account that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq76_HTML.gif in (2.18), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ21_HTML.gif
(2.19)
where here and throughout https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq77_HTML.gif denotes a generic constant that is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq78_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq79_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq80_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq81_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq82_HTML.gif . Taking into account that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq83_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq84_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq85_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ22_HTML.gif
(2.20)
If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq86_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq87_HTML.gif , and we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ23_HTML.gif
(2.21)
Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ24_HTML.gif
(2.22)
In a similar way we can estimate https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq88_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq89_HTML.gif and conclude that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ25_HTML.gif
(2.23)
From here and (2.8), we conclude that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ26_HTML.gif
(2.24)

From (2.3), (2.15), by the mean-value theorem, we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq90_HTML.gif satisfies the difference problem

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ27_HTML.gif
(2.25)

Using the assumption on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq91_HTML.gif from (1.1) and (2.24), by (2.6), we prove the theorem.

2.3. The Monotone Iterative Method

In this section, we construct an iterative method for solving the nonlinear difference scheme (2.3) which possesses monotone convergence.

Additionally, we assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq92_HTML.gif from (1.1) satisfies the two-sided constraint
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ28_HTML.gif
(2.26)
The iterative method is constructed in the following way. Choose an initial mesh function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq93_HTML.gif , then the iterative sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq94_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq95_HTML.gif , is defined by the recurrence formulae
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ29_HTML.gif
(2.27)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq96_HTML.gif is the residual of the difference scheme (2.3) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq97_HTML.gif .

We say that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq98_HTML.gif is an upper solution of (2.3) if it satisfies the inequalities
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ30_HTML.gif
(2.28)
Similarly, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq99_HTML.gif is called a lower solution if it satisfies the reversed inequalities. Upper and lower solutions satisfy the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ31_HTML.gif
(2.29)
Indeed, by the definition of lower and upper solutions and the mean-value theorem, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq100_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ32_HTML.gif
(2.30)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq101_HTML.gif . In view of the maximum principle in Lemma 2.2, we conclude the required inequality.

The following theorem gives the monotone property of the iterative method (2.27).

Theorem 2.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq102_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq103_HTML.gif be upper and lower solutions of (2.3) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq104_HTML.gif satisfy (2.26). Then the upper sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq105_HTML.gif generated by (2.27) converges monotonically from above to the unique solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq106_HTML.gif of (2.3), the lower sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq107_HTML.gif generated by (2.27) converges monotonically from below to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq108_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ33_HTML.gif
(2.31)

and the sequences converge at the linear rate https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq109_HTML.gif .

Proof.

We consider only the case of the upper sequence. If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq110_HTML.gif is an upper solution, then from (2.27) we conclude that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ34_HTML.gif
(2.32)
From Lemma 2.2, by the maximum principle for the difference operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq111_HTML.gif , it follows that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq112_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq113_HTML.gif . Using the mean-value theorem and the equation for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq114_HTML.gif , we represent https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq115_HTML.gif in the form
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ35_HTML.gif
(2.33)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq116_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq117_HTML.gif . Since the mesh function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq118_HTML.gif is nonpositive on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq119_HTML.gif and taking into account (2.26), we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq120_HTML.gif is an upper solution. By induction on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq121_HTML.gif , we obtain that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq122_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq123_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq124_HTML.gif , and prove that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq125_HTML.gif is a monotonically decreasing sequence of upper solutions.

We now prove that the monotone sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq126_HTML.gif converges to the solution of (2.3). Similar to (2.33), we obtain

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ36_HTML.gif
(2.34)
and from (2.27), it follows that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq127_HTML.gif satisfies the difference equation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ37_HTML.gif
(2.35)
Using (2.26) and (2.6), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ38_HTML.gif
(2.36)
This proves the convergence of the upper sequence at the linear rate https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq128_HTML.gif . Now by linearity of the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq129_HTML.gif and the continuity of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq130_HTML.gif , we have also from (2.27) that the mesh function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq131_HTML.gif defined by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ39_HTML.gif
(2.37)
is the exact solution to (2.3). The uniqueness of the solution to (2.3) follows from estimate (2.6). Indeed, if by contradiction, we assume that there exist two solutions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq132_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq133_HTML.gif to (2.3), then by the mean-value theorem, the difference https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq134_HTML.gif satisfies the difference problem
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ40_HTML.gif
(2.38)

By (2.6), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq135_HTML.gif which leads to the uniqueness of the solution to (2.3). This proves the theorem.

Consider the following approach for constructing initial upper and lower solutions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq136_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq137_HTML.gif . Introduce the difference problems
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ41_HTML.gif
(2.39)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq138_HTML.gif from (2.26). Then the functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq139_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq140_HTML.gif are upper and lower solutions, respectively. We check only that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq141_HTML.gif is an upper solution. From the maximum principle in Lemma 2.2, it follows that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq142_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq143_HTML.gif . Now using the difference equation for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq144_HTML.gif and the mean-value theorem, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ42_HTML.gif
(2.40)

Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq145_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq146_HTML.gif is nonnegative, we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq147_HTML.gif is an upper solution.

Theorem 2.5.

If the initial upper or lower solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq148_HTML.gif is chosen in the form of (2.39), then the monotone iterative method (2.27) converges https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq149_HTML.gif -uniformly to the solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq150_HTML.gif of the nonlinear difference scheme (2.3)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ43_HTML.gif
(2.41)

Proof.

From (2.27), (2.39), and the mean-value theorem, by (2.6),
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ44_HTML.gif
(2.42)
From here and estimating https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq151_HTML.gif from (2.39) by (2.6),
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ45_HTML.gif
(2.43)
we conclude the estimate on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq152_HTML.gif in the form
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ46_HTML.gif
(2.44)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq153_HTML.gif is defined in the theorem. From here and (2.36), we conclude that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ47_HTML.gif
(2.45)
Using this estimate, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ48_HTML.gif
(2.46)

Taking into account that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq154_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq155_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq156_HTML.gif is the solution to (2.3), we conclude the theorem.

From Theorems 2.3 and 2.5 we conclude the following theorem.

Theorem 2.6.

Suppose that the initial upper or lower solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq157_HTML.gif is chosen in the form of (2.39). Then the monotone iterative method (2.27) on the piecewise uniform mesh (2.8) converges https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq158_HTML.gif -uniformly to the solution of problem (1.1):
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ49_HTML.gif
(2.47)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq159_HTML.gif and constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq160_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq161_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq162_HTML.gif .

3. The Parabolic Problem

3.1. The Nonlinear Difference Scheme

Introduce uniform mesh https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq163_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq164_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ50_HTML.gif
(3.1)
For approximation of problem (1.2), we use the implicit difference scheme
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ51_HTML.gif
(3.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq165_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq166_HTML.gif are defined in (2.2) and (2.3), respectively. We introduce the linear version of problem (3.2)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ52_HTML.gif
(3.3)

We now formulate a discrete maximum principle for the difference operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq167_HTML.gif and give an estimate of the solution to (3.3).

Lemma 3.1.
  1. (i)
    If a mesh function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq168_HTML.gif on a time level https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq169_HTML.gif satisfies the conditions
    https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ53_HTML.gif
    (3.4)
     
then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq170_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq171_HTML.gif .
  1. (ii)

    If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq172_HTML.gif , then the following estimate of the solution to (3.3) holds true:

     
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ54_HTML.gif
(3.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq173_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq174_HTML.gif .

The proof of the lemma can be found in [6].

3.2. The Monotone Iterative Method

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq175_HTML.gif from (3.2) satisfies the two-sided constraint
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ55_HTML.gif
(3.6)
We consider the following iterative method for solving (3.2). Choose an initial mesh function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq176_HTML.gif . On each time level, the iterative sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq177_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq178_HTML.gif , is defined by the recurrence formulae
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ56_HTML.gif
(3.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq179_HTML.gif is the residual of the difference scheme (3.2) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq180_HTML.gif .

On a time level https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq181_HTML.gif , we say that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq182_HTML.gif is an upper solution of (3.2) with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq183_HTML.gif if it satisfies the inequalities
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ57_HTML.gif
(3.8)
Similarly, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq184_HTML.gif is called a lower solution if it satisfies all the reversed inequalities. Upper and lower solutions satisfy the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ58_HTML.gif
(3.9)

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq185_HTML.gif satisfies (3.6). Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq186_HTML.gif be given and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq187_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq188_HTML.gif be upper and lower solutions of (3.2) corresponding https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq189_HTML.gif . Then the upper sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq190_HTML.gif generated by (3.7) converges monotonically from above to the unique solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq191_HTML.gif of the problem
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ59_HTML.gif
(3.10)
the lower sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq192_HTML.gif generated by (3.7) converges monotonically from below to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq193_HTML.gif and the following inequalities hold
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ60_HTML.gif
(3.11)

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq194_HTML.gif is an upper solution, then from (3.7) we conclude that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ61_HTML.gif
(3.12)
From Lemma 3.1, it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ62_HTML.gif
(3.13)

and from (3.7), it follows that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq195_HTML.gif satisfies the boundary conditions.

Using the mean-value theorem and the equation for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq196_HTML.gif from (3.7), we represent https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq197_HTML.gif in the form

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ63_HTML.gif
(3.14)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq198_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq199_HTML.gif . Since the mesh function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq200_HTML.gif is nonpositive on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq201_HTML.gif and taking into account (3.6), we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq202_HTML.gif is an upper solution to (3.2). By induction on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq203_HTML.gif , we obtain that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq204_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq205_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq206_HTML.gif , and prove that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq207_HTML.gif is a monotonically decreasing sequence of upper solutions.

We now prove that the monotone sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq208_HTML.gif converges to the solution of (3.2). The sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq209_HTML.gif is monotonically decreasing and bounded below by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq210_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq211_HTML.gif is any lower solution (3.9). Now by linearity of the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq212_HTML.gif and the continuity of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq213_HTML.gif , we have also from (3.7) that the mesh function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq214_HTML.gif defined by

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ64_HTML.gif
(3.15)
is an exact solution to (3.2). If by contradiction, we assume that there exist two solutions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq215_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq216_HTML.gif to (3.2), then by the mean-value theorem, the difference https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq217_HTML.gif satisfies the system
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ65_HTML.gif
(3.16)

By Lemma 3.1, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq218_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq219_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq220_HTML.gif . Introduce the difference problems
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ66_HTML.gif
(3.17)

The functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq221_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq222_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq223_HTML.gif satisfy (3.6). Suppose that on each time level the number of iterates https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq224_HTML.gif . Then for the monotone iterative methods (3.7), the following estimate on convergence rate holds:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ67_HTML.gif
(3.18)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq225_HTML.gif is the solution to (3.2), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq226_HTML.gif , and constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq227_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq228_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq229_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq230_HTML.gif .

Proof.

Similar to (3.14), using the mean-value theorem and the equation for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq231_HTML.gif from (3.7), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ68_HTML.gif
(3.19)
From here and (3.7), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ69_HTML.gif
(3.20)
Using (3.5) and (3.6), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ70_HTML.gif
(3.21)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq232_HTML.gif is defined in (3.18).

Introduce the notation

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ71_HTML.gif
(3.22)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq233_HTML.gif . Using the mean-value theorem, from (3.2) and (3.19), we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq234_HTML.gif satisfies the problem
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ72_HTML.gif
(3.23)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq235_HTML.gif , and we have taken into account that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq236_HTML.gif . By (3.5), (3.6), and (3.21),
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ73_HTML.gif
(3.24)

Using (3.6), (3.17), and the mean-value theorem, estimate https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq237_HTML.gif from (3.7) by (3.5),

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ74_HTML.gif
(3.25)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq238_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq239_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq240_HTML.gif ), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq241_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq242_HTML.gif . Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ75_HTML.gif
(3.26)

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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ76_HTML.gif
(3.27)
Using (3.21), by (3.5),
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ77_HTML.gif
(3.28)
Using (3.17), estimate https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq243_HTML.gif from (3.7) by (3.5),
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ78_HTML.gif
(3.29)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq244_HTML.gif . As follows from Theorem 3.2, the monotone sequences https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq245_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq246_HTML.gif are bounded from above and below by, respectively, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq247_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq248_HTML.gif . Applying (3.5) to problem (3.17) at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq249_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ79_HTML.gif
(3.30)
where constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq250_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq251_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq252_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq253_HTML.gif . Thus, we prove that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq254_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq255_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq256_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq257_HTML.gif . From (3.26) and (3.28), we conclude
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ80_HTML.gif
(3.31)
By induction on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq258_HTML.gif , we prove
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ81_HTML.gif
(3.32)

where all constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq259_HTML.gif are independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq260_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq261_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq262_HTML.gif . Taking into account that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq263_HTML.gif , we prove the estimate (3.18) with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq264_HTML.gif .

In [4], we prove that the difference scheme (3.2) on the piecewise uniform mesh (2.8) converges https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq265_HTML.gif -uniformly to the solution of problem (1.2):
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ82_HTML.gif
(3.33)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq266_HTML.gif is the exact solution to (3.2), and constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq267_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq268_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq269_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq270_HTML.gif . 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq271_HTML.gif is chosen in the form of (3.17) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq272_HTML.gif . Then the monotone iterative method (3.7) on the piecewise uniform mesh (2.8) converges https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq273_HTML.gif -uniformly to the solution of problem (1.2):
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ83_HTML.gif
(3.34)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq274_HTML.gif , and constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq275_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq276_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq277_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq278_HTML.gif .

4. Numerical Experiments

It is found that in all numerical experiments the basic feature of monotone convergence of the upper and lower sequences is observed. In fact, the monotone property of the sequences holds at every mesh point in the domain. This is, of course, to be expected from the analytical consideration.

4.1. The Elliptic Problem

Consider problem (1.1) with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq279_HTML.gif . We mention that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq280_HTML.gif is the solution of the reduced problem, where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq281_HTML.gif . This problem gives https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq282_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq283_HTML.gif , and initial lower and upper solutions are chosen in the form of (2.39). The stopping criterion for the monotone iterative method (2.27) is
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ84_HTML.gif
(4.1)

Our numerical experiments show that for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq284_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq285_HTML.gif , iteration counts for monotone method (2.27) on the piecewise uniform mesh are independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq286_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq287_HTML.gif , and equals 12 and 8 for the lower and upper sequences, respectively. These numerical results confirm our theoretical results stated in Theorem 2.5.

In Table 1, we present numbers of iterations for solving the test problem by the Newton iterative method with the initial iterations https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq288_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq289_HTML.gif . Here https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq290_HTML.gif is in use, and we denote by an "*" if more than 100 iterations is needed to satisfy the stopping criterion, or if the method diverges. The numerical results indicate that the Newton method cannot be used successfully for this test problem.
Table 1

Numbers of iterations for the Newton iterative method.

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq291_HTML.gif

128

256

512

1024

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq292_HTML.gif

7

7

9

*

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq293_HTML.gif

8

11

18

*

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq294_HTML.gif

73

*

*

*

4.2. The Parabolic Problem

For the parabolic problem (1.2), we consider the test problem with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq295_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq296_HTML.gif . This problem gives https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq297_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq298_HTML.gif , and the initial lower and upper solutions are chosen in the form of (3.17).

The stopping test for the monotone method (3.7) is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ85_HTML.gif
(4.2)

Our numerical experiments show that for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq299_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq300_HTML.gif , on each time level the number of iterations for monotone method (3.7) on the piecewise uniform mesh is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq301_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq302_HTML.gif and equal 4, 4, and 3 for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq303_HTML.gif , respectively. These numerical results confirm our theoretical results stated in Theorem 3.3.

Authors’ Affiliations

(1)
Institute of Fundamental Sciences, Massey University

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Copyright

© Igor Boglaev. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.