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
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ1_HTML.gif
(1.1)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq1_HTML.gif is a positive parameter, and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq2_HTML.gif is sufficiently smooth function. For http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq4_HTML.gif and http://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
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ2_HTML.gif
(1.2)

where http://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 http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq11_HTML.gif -direction, which converges http://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 http://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 http://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 http://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 http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ3_HTML.gif
(2.1)

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

For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq19_HTML.gif , the boundary layers appear near http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq20_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq22_HTML.gif
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ5_HTML.gif
(2.3)
where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq23_HTML.gif and http://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
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ6_HTML.gif
(2.4)

where http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq27_HTML.gif satisfies the conditions
    http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ7_HTML.gif
    (2.5)
     
then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq28_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq29_HTML.gif .
  1. (ii)

    If http://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:

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

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq31_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq33_HTML.gif into three parts http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq34_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq35_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq36_HTML.gif . Assuming that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq37_HTML.gif is divisible by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq38_HTML.gif , in the parts http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq39_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq40_HTML.gif we use the uniform mesh with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq41_HTML.gif mesh points, and in the part http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq42_HTML.gif the uniform mesh with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq44_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq45_HTML.gif are determined by
http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq48_HTML.gif is coarse, such that
http://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 http://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):
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ11_HTML.gif
(2.9)

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

Proof.

Using Green's function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq53_HTML.gif of the differential operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq54_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq55_HTML.gif , we represent the exact solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq56_HTML.gif in the form
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq57_HTML.gif is given by
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ13_HTML.gif
(2.11)
and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq58_HTML.gif are defined by
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ14_HTML.gif
(2.12)
Equating the derivatives http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq59_HTML.gif and http://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:
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq61_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq62_HTML.gif . Representing http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq63_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq64_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq65_HTML.gif in the forms
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ17_HTML.gif
(2.15)
where the truncation error http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq66_HTML.gif of the exact solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq67_HTML.gif to (1.1) is defined by
http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq68_HTML.gif holds:

http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq69_HTML.gif in (2.17) on the interval http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq70_HTML.gif . Consider the following three cases: http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq71_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq72_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq73_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq74_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq75_HTML.gif , and taking into account that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq76_HTML.gif in (2.18), we have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ21_HTML.gif
(2.19)
where here and throughout http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq78_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq79_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq80_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq81_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq82_HTML.gif . Taking into account that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq83_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq84_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq85_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ22_HTML.gif
(2.20)
If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq86_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq87_HTML.gif , and we have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ23_HTML.gif
(2.21)
Thus,
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq88_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq89_HTML.gif and conclude that
http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq90_HTML.gif satisfies the difference problem

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

Using the assumption on http://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 http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq93_HTML.gif , then the iterative sequence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq94_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq95_HTML.gif , is defined by the recurrence formulae
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ29_HTML.gif
(2.27)

where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq97_HTML.gif .

We say that http://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
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ30_HTML.gif
(2.28)
Similarly, http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq100_HTML.gif we have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ32_HTML.gif
(2.30)

where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq102_HTML.gif , http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq106_HTML.gif of (2.3), the lower sequence http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq108_HTML.gif :
http://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 http://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 http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq111_HTML.gif , it follows that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq112_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq114_HTML.gif , we represent http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq115_HTML.gif in the form
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ35_HTML.gif
(2.33)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq116_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq117_HTML.gif . Since the mesh function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq118_HTML.gif is nonpositive on http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq121_HTML.gif , we obtain that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq122_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq123_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq124_HTML.gif , and prove that http://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 http://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

http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq127_HTML.gif satisfies the difference equation
http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq128_HTML.gif . Now by linearity of the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq129_HTML.gif and the continuity of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq131_HTML.gif defined by
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq132_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq134_HTML.gif satisfies the difference problem
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ40_HTML.gif
(2.38)

By (2.6), http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq136_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq137_HTML.gif . Introduce the difference problems
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ41_HTML.gif
(2.39)
where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq138_HTML.gif from (2.26). Then the functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq139_HTML.gif , http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq142_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq143_HTML.gif . Now using the difference equation for http://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
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ42_HTML.gif
(2.40)

Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq145_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq146_HTML.gif is nonnegative, we conclude that http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq149_HTML.gif -uniformly to the solution http://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)
http://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),
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ44_HTML.gif
(2.42)
From here and estimating http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq151_HTML.gif from (2.39) by (2.6),
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq152_HTML.gif in the form
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ46_HTML.gif
(2.44)
where http://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
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ48_HTML.gif
(2.46)

Taking into account that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq154_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq155_HTML.gif , where http://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 http://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 http://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):
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ49_HTML.gif
(2.47)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq159_HTML.gif and constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq160_HTML.gif is independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq161_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq163_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq164_HTML.gif
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ51_HTML.gif
(3.2)
where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq165_HTML.gif and http://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)
http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq168_HTML.gif on a time level http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq169_HTML.gif satisfies the conditions
    http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ53_HTML.gif
    (3.4)
     
then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq170_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq171_HTML.gif .
  1. (ii)

    If http://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:

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

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq173_HTML.gif , http://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 http://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
http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq177_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq178_HTML.gif , is defined by the recurrence formulae
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ56_HTML.gif
(3.7)

where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq180_HTML.gif .

On a time level http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq181_HTML.gif , we say that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq183_HTML.gif if it satisfies the inequalities
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ57_HTML.gif
(3.8)
Similarly, http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq185_HTML.gif satisfies (3.6). Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq186_HTML.gif be given and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq187_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq189_HTML.gif . Then the upper sequence http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq191_HTML.gif of the problem
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ59_HTML.gif
(3.10)
the lower sequence http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq193_HTML.gif and the following inequalities hold
http://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 http://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

http://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
http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq196_HTML.gif from (3.7), we represent http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq197_HTML.gif in the form

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

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq198_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq199_HTML.gif . Since the mesh function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq200_HTML.gif is nonpositive on http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq203_HTML.gif , we obtain that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq204_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq205_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq206_HTML.gif , and prove that http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq210_HTML.gif , where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq212_HTML.gif and the continuity of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq214_HTML.gif defined by

http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq215_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq217_HTML.gif satisfies the system
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ65_HTML.gif
(3.16)

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

The functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq221_HTML.gif , http://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 http://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 http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ67_HTML.gif
(3.18)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq225_HTML.gif is the solution to (3.2), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq226_HTML.gif , and constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq227_HTML.gif is independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq228_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq229_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq231_HTML.gif from (3.7), we have
http://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
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ70_HTML.gif
(3.21)

where http://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

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ71_HTML.gif
(3.22)
where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq234_HTML.gif satisfies the problem
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ72_HTML.gif
(3.23)
where http://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 http://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),
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq237_HTML.gif from (3.7) by (3.5),

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

http://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),
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ77_HTML.gif
(3.28)
Using (3.17), estimate http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq243_HTML.gif from (3.7) by (3.5),
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ78_HTML.gif
(3.29)
where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq245_HTML.gif and http://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, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq247_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq249_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ79_HTML.gif
(3.30)
where constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq250_HTML.gif is independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq251_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq252_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq253_HTML.gif . Thus, we prove that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq254_HTML.gif is independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq255_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq256_HTML.gif and http://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
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ80_HTML.gif
(3.31)
By induction on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq258_HTML.gif , we prove
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ81_HTML.gif
(3.32)

where all constants http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq259_HTML.gif are independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq260_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq261_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq262_HTML.gif . Taking into account that http://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 http://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 http://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):
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ82_HTML.gif
(3.33)

where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq267_HTML.gif is independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq268_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq269_HTML.gif and http://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 http://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 http://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 http://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):
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_Equ83_HTML.gif
(3.34)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq274_HTML.gif , and constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq275_HTML.gif is independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq276_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq277_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq279_HTML.gif . We mention that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq281_HTML.gif . This problem gives http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq282_HTML.gif , http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq284_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq286_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq288_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq289_HTML.gif . Here http://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.

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

128

256

512

1024

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

7

7

9

*

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

8

11

18

*

http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq295_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq296_HTML.gif . This problem gives http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq297_HTML.gif , http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq299_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F320606/MediaObjects/13661_2009_Article_840_IEq301_HTML.gif and http://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 http://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.