Boundary Value Problems

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Robust Monotone Iterates for Nonlinear Singularly Perturbed Boundary Value Problems

Boundary Value Problems20092009:320606

DOI: 10.1155/2009/320606

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
(1.1)

where is a positive parameter, and is sufficiently smooth function. For this problem is singularly perturbed, and the solution has boundary layers near and (see [1] for details).

The second problem is the one-dimensional parabolic problem
(1.2)

where is a positive parameter. Under suitable continuity and compatibility conditions on the data, a unique solution of this problem exists. For problem (1.2) is singularly perturbed and has boundary layers near the lateral boundary of (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 -uniform numerical method for solving problem (1.1), that is, a numerical method which generates -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 -direction, which converges -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 -uniformly convergent monotone iterative methods for solving -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 -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 is the solution to (1.1), the following estimates hold:
(2.1)

where constant is independent of .

For , the boundary layers appear near and .

2.1. The Nonlinear Difference Scheme

Introduce a nonuniform mesh
(2.2)
For solving (1.1), we use the classical difference scheme
(2.3)
where and . We introduce the linear version of this problem
(2.4)

where . We now formulate a discrete maximum principle for the difference operator and give an estimate of the solution to (2.4).

Lemma 2.2.
1. (i)
If a mesh function satisfies the conditions
(2.5)

then , .
1. (ii)

If , then the following estimate of the solution to (2.4) holds true:

(2.6)

where , .

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 into three parts , and . Assuming that is divisible by , in the parts , we use the uniform mesh with mesh points, and in the part the uniform mesh with mesh points is in use. The transition points and are determined by
(2.7)
This defines the piecewise uniform mesh. If the parameter is small enough, then the uniform mesh inside of the boundary layers with the step size is fine, and the uniform mesh outside of the boundary layers with the step size is coarse, such that
(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 -uniformly to the solution of (1.1):
(2.9)

where constant is independent of and .

Proof.

Using Green's function of the differential operator on , we represent the exact solution in the form
(2.10)
where the local Green function is given by
(2.11)
and are defined by
(2.12)
Equating the derivatives and , we get the following integral-difference formula:
(2.13)
where here and below we suppress variable in . Representing on and in the forms
(2.14)
the above integral-difference formula can be written as
(2.15)
where the truncation error of the exact solution to (1.1) is defined by
(2.16)
From here, it follows that
(2.17)

From Lemma 2.1, the following estimate on holds:

(2.18)
We estimate the truncation error in (2.17) on the interval . Consider the following three cases: , and . If , then , and taking into account that in (2.18), we have
(2.19)
where here and throughout denotes a generic constant that is independent of and . If , then , . Taking into account that , and , we have
(2.20)
If , then , and we have
(2.21)
Thus,
(2.22)
In a similar way we can estimate on and conclude that
(2.23)
From here and (2.8), we conclude that
(2.24)

From (2.3), (2.15), by the mean-value theorem, we conclude that satisfies the difference problem

(2.25)

Using the assumption on 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 from (1.1) satisfies the two-sided constraint
(2.26)
The iterative method is constructed in the following way. Choose an initial mesh function , then the iterative sequence , , is defined by the recurrence formulae
(2.27)

where is the residual of the difference scheme (2.3) on .

We say that is an upper solution of (2.3) if it satisfies the inequalities
(2.28)
Similarly, is called a lower solution if it satisfies the reversed inequalities. Upper and lower solutions satisfy the inequality
(2.29)
Indeed, by the definition of lower and upper solutions and the mean-value theorem, for we have
(2.30)

where . 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 , be upper and lower solutions of (2.3) and satisfy (2.26). Then the upper sequence generated by (2.27) converges monotonically from above to the unique solution of (2.3), the lower sequence generated by (2.27) converges monotonically from below to :
(2.31)

and the sequences converge at the linear rate .

Proof.

We consider only the case of the upper sequence. If is an upper solution, then from (2.27) we conclude that
(2.32)
From Lemma 2.2, by the maximum principle for the difference operator , it follows that , . Using the mean-value theorem and the equation for , we represent in the form
(2.33)

where , . Since the mesh function is nonpositive on and taking into account (2.26), we conclude that is an upper solution. By induction on , we obtain that , , , and prove that is a monotonically decreasing sequence of upper solutions.

We now prove that the monotone sequence converges to the solution of (2.3). Similar to (2.33), we obtain

(2.34)
and from (2.27), it follows that satisfies the difference equation
(2.35)
Using (2.26) and (2.6), we have
(2.36)
This proves the convergence of the upper sequence at the linear rate . Now by linearity of the operator and the continuity of , we have also from (2.27) that the mesh function defined by
(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 and to (2.3), then by the mean-value theorem, the difference satisfies the difference problem
(2.38)

By (2.6), 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 and . Introduce the difference problems
(2.39)
where from (2.26). Then the functions , are upper and lower solutions, respectively. We check only that is an upper solution. From the maximum principle in Lemma 2.2, it follows that on . Now using the difference equation for and the mean-value theorem, we have
(2.40)

Since and is nonnegative, we conclude that is an upper solution.

Theorem 2.5.

If the initial upper or lower solution is chosen in the form of (2.39), then the monotone iterative method (2.27) converges -uniformly to the solution of the nonlinear difference scheme (2.3)
(2.41)

Proof.

From (2.27), (2.39), and the mean-value theorem, by (2.6),
(2.42)
From here and estimating from (2.39) by (2.6),
(2.43)
we conclude the estimate on in the form
(2.44)
where is defined in the theorem. From here and (2.36), we conclude that
(2.45)
Using this estimate, we have
(2.46)

Taking into account that as , where 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 is chosen in the form of (2.39). Then the monotone iterative method (2.27) on the piecewise uniform mesh (2.8) converges -uniformly to the solution of problem (1.1):
(2.47)

where and constant is independent of and .

3. The Parabolic Problem

3.1. The Nonlinear Difference Scheme

Introduce uniform mesh on
(3.1)
For approximation of problem (1.2), we use the implicit difference scheme
(3.2)
where and are defined in (2.2) and (2.3), respectively. We introduce the linear version of problem (3.2)
(3.3)

We now formulate a discrete maximum principle for the difference operator and give an estimate of the solution to (3.3).

Lemma 3.1.
1. (i)
If a mesh function on a time level satisfies the conditions
(3.4)

then , .
1. (ii)

If , then the following estimate of the solution to (3.3) holds true:

(3.5)

where , .

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

3.2. The Monotone Iterative Method

Assume that from (3.2) satisfies the two-sided constraint
(3.6)
We consider the following iterative method for solving (3.2). Choose an initial mesh function . On each time level, the iterative sequence , , is defined by the recurrence formulae
(3.7)

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

On a time level , we say that is an upper solution of (3.2) with respect to if it satisfies the inequalities
(3.8)
Similarly, is called a lower solution if it satisfies all the reversed inequalities. Upper and lower solutions satisfy the inequality
(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 satisfies (3.6). Let be given and , be upper and lower solutions of (3.2) corresponding . Then the upper sequence generated by (3.7) converges monotonically from above to the unique solution of the problem
(3.10)
the lower sequence generated by (3.7) converges monotonically from below to and the following inequalities hold
(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 is an upper solution, then from (3.7) we conclude that

(3.12)
From Lemma 3.1, it follows that
(3.13)

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

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

(3.14)

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

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

(3.15)
is an exact solution to (3.2). If by contradiction, we assume that there exist two solutions and to (3.2), then by the mean-value theorem, the difference satisfies the system
(3.16)

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

Consider the following approach for constructing initial upper and lower solutions and . Introduce the difference problems
(3.17)

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

Theorem 3.3.

Let initial upper or lower solution be chosen in the form of (3.17), and let satisfy (3.6). Suppose that on each time level the number of iterates . Then for the monotone iterative methods (3.7), the following estimate on convergence rate holds:
(3.18)

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

Proof.

Similar to (3.14), using the mean-value theorem and the equation for from (3.7), we have
(3.19)
From here and (3.7), we have
(3.20)
Using (3.5) and (3.6), we have
(3.21)

where is defined in (3.18).

Introduce the notation

(3.22)
where . Using the mean-value theorem, from (3.2) and (3.19), we conclude that satisfies the problem
(3.23)
where , and we have taken into account that . By (3.5), (3.6), and (3.21),
(3.24)

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

(3.25)
where is independent of ( ), and . Thus,
(3.26)

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

(3.27)
Using (3.21), by (3.5),
(3.28)
Using (3.17), estimate from (3.7) by (3.5),
(3.29)
where . As follows from Theorem 3.2, the monotone sequences and are bounded from above and below by, respectively, and . Applying (3.5) to problem (3.17) at , we have
(3.30)
where constant is independent of , and . Thus, we prove that is independent of , and . From (3.26) and (3.28), we conclude
(3.31)
By induction on , we prove
(3.32)

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

In [4], we prove that the difference scheme (3.2) on the piecewise uniform mesh (2.8) converges -uniformly to the solution of problem (1.2):
(3.33)

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

Theorem 3.4.

Suppose that on each time level the initial upper or lower solution is chosen in the form of (3.17) and . Then the monotone iterative method (3.7) on the piecewise uniform mesh (2.8) converges -uniformly to the solution of problem (1.2):
(3.34)

where , and constant is independent of , and .

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 . We mention that is the solution of the reduced problem, where . This problem gives , , 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
(4.1)

Our numerical experiments show that for and , iteration counts for monotone method (2.27) on the piecewise uniform mesh are independent of and , 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 , . Here 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.

4.2. The Parabolic Problem

For the parabolic problem (1.2), we consider the test problem with and . This problem gives , , 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
(4.2)

Our numerical experiments show that for and , on each time level the number of iterations for monotone method (3.7) on the piecewise uniform mesh is independent of and and equal 4, 4, and 3 for , 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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