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Robust monotone iterates for solving nonlinear difference scheme are constructed.. In the study of numerical methods for nonlinear singularly perturbed problems, the two major points to b

Volume 2009, Article ID 320606, 17 pages

doi:10.1155/2009/320606

Research Article

Robust Monotone Iterates for Nonlinear Singularly Perturbed Boundary Value Problems

Igor Boglaev

Institute of Fundamental Sciences, Massey University, Private Bag 11-222,

4442 Palmerston North, New Zealand

Correspondence should be addressed to Igor Boglaev,i.boglaev@massey.ac.nz

Received 8 April 2009; Accepted 11 May 2009

Recommended by Donal O’Regan

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

Copyrightq 2009 Igor Boglaev 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

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

−μ2u fx, u  0, x ∈ ω  0, 1, u0  0, u1  0,

f u ≥ c∗ const > 0, x, u ∈ ω × −∞, ∞, f u  ∂f/∂u, 1.1 where μ is a positive parameter, and f is sufficiently smooth function For μ  1 this problem

is singularly perturbed, and the solution has boundary layers near x  0 and x  1 see 1 for details

The second problem is the one-dimensional parabolic problem

−μ2u xx  u t  fx, t, u  0, x, t ∈ Q  ω × 0, T , ω  0, 1,

u 0, t  0, u1, t  0, ux, 0  u0x, x ∈ ω,

f u ≥ 0, x, t, u ∈ Q × −∞, ∞,

1.2

where μ is a positive parameter Under suitable continuity and compatibility conditions on the data, a unique solution of this problem exists For μ  1 problem 1.2 is singularly perturbed and has boundary layers near the lateral boundary ofQ 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 μ-uniform numerical method for solving problem 1.1, that

is, a numerical method which generates μ-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-type3 For solving problem 1.2, we use the implicit difference scheme based on the piecewise uniform mesh in the x-direction, which converges

μ-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 orderingsee 5, Chapter 13 for details

The purpose of this paper is to construct μ-uniformly convergent monotone iterative methods for solving μ-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 μ-uniformly to the solution

of problem1.1 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 from1 contains necessary estimates of the solution to 1.1

Lemma 2.1 If ux ∈ C0ω ∩ C2ω is the solution to 1.1, the following estimates hold:

max

x∈ω |ux| ≤ c−1

∗ max

x∈ω

f x, 0, ux ≤ C1  μ−1Πx,

Πx  exp



−

√

c∗

μ



 exp



−

√

c∗1 − x

μ



,

2.1

where constant C is independent of μ.

For μ  1, the boundary layers appear near x  0 and x  1.

2.1 The Nonlinear Difference Scheme

Introduce a nonuniform mesh ω h

ω h  {x i , 0 ≤ i ≤ N; x0 0, x N  1; h i  x i1 − x i }. 2.2 For solving1.1, we use the classical difference scheme

Lh v x  fx, v  0, x ∈ ω h , v 0  0, v1  0,

Lh v i  −μ2i−1v i1 − v i h i−1− v i − v i−1 h i−1−1, 2.3

where v i  vx i and i  h i−1  h i /2 We introduce the linear version of this problem



Lh  cw x  f0x, x ∈ ω h , w 0  0, w1  0, 2.4

where cx ≥ 0 We now formulate a discrete maximum principle for the difference operator

Lh  c and give an estimate of the solution to 2.4

Lemma 2.2 i If a mesh function wx satisfies the conditions



Lh  cw x ≥ 0 ≤ 0, x ∈ ω h , w 0, w1 ≥ 0 ≤ 0, 2.5

then wx ≥ 0 ≤ 0, x ∈ ω h

ii If cx ≥ c∗ const > 0, then the following estimate of the solution to 2.4 holds true:

w ω h ≤ max f0ω h /c∗, 2.6

where w ω h  maxx∈ω h |wx|, f0ω h  maxx∈ω h |f0x|.

The proof of the lemma can be found in6

2.2 Uniform Convergence on the Piecewise Uniform Mesh

We employ a layer-adapted mesh of a piecewise uniform type3 The piecewise uniform mesh is formed in the following manner We divide the intervalω  0, 1 into three parts

0, ς , ς, 1− ς , and 1− ς, 1 Assuming that N is divisible by 4, in the parts 0, ς , 1− ς, 1 we use the uniform mesh with N/4  1 mesh points, and in the part ς, 1 − ς the uniform mesh with N/2  1 mesh points is in use The transition points ς and 1 − ς are determined by

ς  min

4−1, μ ln N√

This defines the piecewise uniform mesh If the parameter μ is small enough, then the uniform mesh inside of the boundary layers with the step size h μ  4ςN−1 is fine, and the

uniform mesh outside of the boundary layers with the step size h  21 − 2ςN−1is coarse, such that

N−1< h < 2N−1, h μ  4μ√

c∗N−1

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

μ-uniformly to the solution of 1.1:

max

x∈ω h |vx − ux| ≤ CN−1ln N, 2.9

where constant C is independent of μ and N.

Proof Using Green’s function G i of the differential operator μ2d2/dx2 on x i , x i1 , we

represent the exact solution ux in the form

u x  ux i φ I

i x  ux i1 φ II

i x  x i1

x i

G i x, sfs, uds,

2.10

where the local Green function G iis given by

G i x, s  1

μ2w i s

⎧

⎨

⎩

φ I i sφ II

i x, x ≤ s,

φ I i xφ II

i s, x ≥ s,

w i s  φ II

i sφ I i x

xs − φ I

i sφ II i x

xs ,

2.11

and φ I

i x, φ II

i x are defined by

φ I i x  x i1 − x

h i , φ i II x  x − x i

h i , x i ≤ x ≤ x i1 2.12

Equating the derivatives dux i − 0/dx and dux i  0/dx, we get the following

integral-difference formula:

Lh u x i  1

i

x i

x

φ i−1 II sfsds 1

i

x i1

x

φ i I sfsds, 2.13

where here and below we suppress variable u in f Representing fx on x i−1 , x i and

x i , x i1 in the forms

f x  fx i− 0  x

x i

df

ds ds, f x  fx i 0  x

x i

df

ds ds, 2.14 the above integral-difference formula can be written as

Lh u x  fx, u  Trx, x ∈ ω h , 2.15 where the truncation error Trx of the exact solution ux to 1.1 is defined by

Trxi ≡ −1

i

x i

x i−1

φ II i−1 s s

x i

df

dξ dξ



ds −1i

x i1

x i

φ I i s s

x i

df

dξ dξ



ds. 2.16

From here, it follows that

|Trx i| ≤ 1

i

x i

x i−1

φ i−1 II s x i

x i−1



df dξdξ



ds 1i

x i1

x i

φ i I s x i1

x i



df dξdξ



ds. 2.17

FromLemma 2.1, the following estimate on df/dx holds:



dx df ≤ C1 μ−1Πx. 2.18

We estimate the truncation error Tr in2.17 on the interval 0, 1/2 Consider the following three cases: x i ∈ 0, ς, x i  ς, and x i ∈ ς, 1/2 If x i ∈ 0, ς, then h i−1  h i  h μ, and taking into account thatΠx < 2 in 2.18, we have

|Trx i | ≤ Ch μ



1 2μ−1

, x i ∈ 0, ς, 2.19

where here and throughout C denotes a generic constant that is independent of μ and N.

If x i  ς, then h i−1  h μ , h i  h Taking into account that ς  μ ln N/√c∗,Πx < 2, and Πx ≤ 2 exp−√c∗x/μ, we have

|Trς| ≤ C

h μ  h



h2μ

1 2μ−1

 h2 2√

c∗N−1

≤ Ch μ



1 2μ−1

 h  2√

c∗N−1

.

2.20

If x i ∈ ς, 1/2 , then h i−1  h i  h, and we have

|Trx i | ≤ Ch  2√

c∗N−1

, x i ∈ ς, 1/2 2.21

|Trx i | ≤ Ch μ



1 2μ−1

 h  2√

c∗N−1

, x i ∈ 0, 1/2 2.22

In a similar way we can estimate Tr on1/2, 1 and conclude that

|Trx i | ≤ Ch μ



1 2μ−1

 h  2√

c∗N−1

, x i ∈ ω h 2.23 From here and2.8, we conclude that

max

x i ∈ω h |Trx i | ≤ CN−1ln N. 2.24

From2.3, 2.15, by the mean-value theorem, we conclude that w  v − u satisfies the

difference problem

Lh w x  f u w x  −Trx, x ∈ ω h , w 0  w1  0. 2.25

Using the assumption on f ufrom1.1 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 fx, u from 1.1 satisfies the two-sided constraint

0 < c∗≤ f u ≤ c∗, c∗, c∗ const. 2.26

The iterative method is constructed in the following way Choose an initial mesh

function v0, then the iterative sequence{v n }, n ≥ 1, is defined by the recurrence formulae



Lh  c∗

z n x  −R h

x, v n−1

, x ∈ ω h ,

z10  −v00, z11  −v01, z n 0  z n 1  0, n ≥ 2,

v n x  v n−1 x  z n x, x ∈ ω h

,

Rh

x, v n−1

 Lh v n−1 x  fx, v n−1

,

2.27

whereRh x, v n−1 is the residual of the difference scheme 2.3 on v n−1

We say thatvx is an upper solution of 2.3 if it satisfies the inequalities

Lh v x  fx, v ≥ 0, x ∈ ω h , v 0, v1 ≥ 0. 2.28

Similarly, vx is called a lower solution if it satisfies the reversed inequalities Upper and

lower solutions satisfy the inequality

v x ≤ vx, x ∈ ω h

Indeed, by the definition of lower and upper solutions and the mean-value theorem, for δv 

v − v we have

Lh δv  f u xδvx ≥ 0, x ∈ ω h , δv x ≥ 0, x  0, 1, 2.30

where f u x  c u x, vx  ϑxδvx , 0 < ϑx < 1 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 method2.27

Theorem 2.4 Let v0, v0 be upper and lower solutions of 2.3 and f satisfy 2.26 Then the

upper sequence {v n } generated by 2.27 converges monotonically from above to the unique solution

v of 2.3, the lower sequence {v n } generated by 2.27 converges monotonically from below to v:

v n x ≤ v n1 x ≤ vx ≤ v n1 x ≤ v n x, x ∈ ω h

and the sequences converge at the linear rate q  1 − c∗/c∗.

Proof We consider only the case of the upper sequence If v0is an upper solution, then from

2.27 we conclude that



Lh  c∗

z1x ≤ 0, x ∈ ω h , z10, z11 ≤ 0. 2.32

FromLemma 2.2, by the maximum principle for the difference operator Lh c∗, it follows that

z1x ≤ 0, x ∈ ω h

Using the mean-value theorem and the equation for z1, we represent

Rh x, v1 in the form

Rh

x, v1

 −c∗− f u1xz1x, x ∈ ω h , 2.33

where f u1x  f u x, v0x  ϑ1xz1x , 0 < ϑ1x < 1 Since the mesh function z1is

nonpositive on ω hand taking into account2.26, we conclude that v1is an upper solution

By induction on n, we obtain that z n x ≤ 0, x ∈ ω h

, n ≥ 1, and prove that {v n} is a monotonically decreasing sequence of upper solutions

We now prove that the monotone sequence{v n} converges to the solution of 2.3 Similar to2.33, we obtain

Rx, v n−1

 −c∗− f u n−1 xz n−1 x, x ∈ ω h , n ≥ 2, 2.34

and from2.27, it follows that z n1satisfies the difference equation

L  c∗z n x c∗− f u n−1 xz n−1 x, x ∈ ω h 2.35 Using2.26 and 2.6, we have

z n ω h ≤ q n−1 z1ω h 2.36

This proves the convergence of the upper sequence at the linear rate q Now by linearity of

the operatorLh and the continuity of f, we have also from 2.27 that the mesh function v

defined by

v x  lim

is the exact solution to2.3 The uniqueness of the solution to 2.3 follows from estimate

2.6 Indeed, if by contradiction, we assume that there exist two solutions v1and v2to2.3, then by the mean-value theorem, the difference δv  v1− v2satisfies the difference problem

Lh δv  f u δv  0, x ∈ ω h , δv 0  δv1  0. 2.38

By 2.6, δv  0 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

v0and v0 Introduce the difference problems



Lh  c∗



v0ν  νf x, 0, x ∈ ω h ,

v ν00  v ν01  0, ν  1, −1,

2.39

where c∗from2.26 Then the functions v01 , v0−1 are upper and lower solutions, respectively

We check only that v01 is an upper solution From the maximum principle inLemma 2.2, it

follows that v10 ≥ 0 on ω h Now using the difference equation for v10and the mean-value theorem, we have

Rh

x, v01 

 fx, 0 f x, 0  f0

u − c∗ v01 . 2.40

Since f u0≥ c∗and v10is nonnegative, we conclude that v01 is an upper solution

Theorem 2.5 If the initial upper or lower solution v0 is chosen in the form of 2.39, then the

monotone iterative method2.27 converges μ-uniformly to the solution v of the nonlinear difference

scheme2.3



v n − v

ω h ≤ c0q n

1− qfx, 0

ω h ,

q  1 − c∗/c∗< 1, c0 3c∗ c∗/c∗c∗.

2.41

Proof From2.27, 2.39, and the mean-value theorem, by 2.6,



z1

c∗



Lh v0

c∗



fx, v0

ω h

≤ 1

c∗



c∗v0

ω hfx, 0

ω h



 1

c∗fx, 0

ω h v0

ω h

2.42

From here and estimating v0from2.39 by 2.6,



v0

c∗

fx, 0

we conclude the estimate on z1in the form



z1

ω h ≤ c0fx, 0

where c0is defined in the theorem From here and2.36, we conclude that



z n

ω h ≤ c0q n−1fx, 0

Using this estimate, we have



v nk − v n

ω h ≤nk−1

in



v i1 − v i

ω h nk−1

in



z i1

ω h

≤ q

1− qz n

ω h ≤ c0q n

1− qfx, 0

ω h

2.46

Taking into account that lim v nk  v as k → ∞, where v is the solution to 2.3, we conclude the theorem

From Theorems2.3and2.5we conclude the following theorem

Theorem 2.6 Suppose that the initial upper or lower solution v0is chosen in the form of 2.39.

Then the monotone iterative method2.27 on the piecewise uniform mesh 2.8 converges μ-uniformly

to the solution of problem1.1:



v n − u

ω h ≤ CN−1ln N  q n

where q  1 − c∗/c∗, and constant C is independent of μ and N.

3 The Parabolic Problem

3.1 The Nonlinear Difference Scheme

Introduce uniform mesh ω τon0, T

ω τ  {t k  kτ, 0 ≤ k ≤ N τ , N τ τ  T }. 3.1

For approximation of problem1.2, we use the implicit difference scheme

Lvx, t − τ−1v x, t − τ  −fx, t, v, x, t ∈ ω h × ω τ \ {∅},

v 0, t  0, v1, t  0, vx, 0  u0x, x ∈ ω h

,

L  Lh  τ−1,

3.2

where ω handLhare defined in2.2 and 2.3, respectively We introduce the linear version

of problem3.2

L  cwx, t  f0x, t, x ∈ ω h ,

w 0, t  0, w1, t  0, cx, t ≥ 0, x ∈ ω h

.

3.3

We now formulate a discrete maximum principle for the difference operator L  c and give

an estimate of the solution to3.3

Lemma 3.1 i If a mesh function wx, t on a time level t ∈ ω τ \ {∅} satisfies the conditions

L  cwx, t ≥ 0 ≤ 0, x ∈ ω h , w 0, t, w1, t ≥ 0 ≤ 0, 3.4

then wx, t ≥ 0 ≤ 0, x ∈ ω h

Theorem 2.5 If the initial upper or lower solution v0 is chosen in the form... i ∈ ς, 1/2 2.21

|Trx i | ≤ Ch μ... h , v 0, v1 ≥ 0. 2.28

Similarly, vx is called a lower solution if it satisfies