A posteriori error estimates for one-dimensional convection-diffussion problems

(1)

ELSEVIER

An Intemational Journal Available online at www.sciencedirect.com

computers &

• =,"-=" @ o , - " = ~ .

m a t h e m a t i c s

with applications Computers and Mathematics with Applications 5t (2006) 915-926

www.elsevier.com/locate/camwa

A Posteriori E r r o r E s t i m a t e s

for O n e - D i m e n s i o n a l

C o n v e c t i o n - D i f f u s s i o n P r o b l e m s

R . V U L A N O V I C AND G . H O V H A N N I S Y A N

Department of Mathematical Sciences Kent State University Stark Campus

6000 Frank Ave. NW Canton, OH 44720-7599, U.S.A.

<rvulanovic><ghovhannisyan>©st ark. kent. edu

(Received January 2005; revised and accepted November 2005)

A b s t r a c t - - T h i s paper is concerned with the upwind finite-difference discretization of a quasilinear singularly perturbed boundary value problem without turning points. Kopteva's

a posteriori

Keywords--Convection-diffusion, Quasilinear boundary value problem, Singular perturbation, Finite-difference scheme, Bakhvalov mesh, Shishkin mesh,

A posteriori

error estimate.

1. I N T R O D U C T I O N

We consider the singularly perturbed quasilinear convection-diffusion problem,

T u : = - ¢ u " - b ( x , u ) ' + c ( x , u ) = O ,

for x e X : = [0,1], u(0) = u(1) = 0, (1) where ~ is the p e r t u r b a t i o n parameter, 0 < s << 1, a n d b a n d c are two

C 2 ( X x R)

functions satisfying

bu(x,u)>~>0, cu(x,u)>~,

x C Z , u 6 R , (2)

where 3' <- 0. Since we assume t h a t E is small enough, it follows t h a t f12 +4gU > 0 a n d t h e n by [2], problem (1) has a u n i q u e solution

uE E Ca(X).

This solution in general exhibits a b o u n d a r y layer of e x p o n e n t i a l type near x = 0 a n d its derivatives can be e s t i m a t e d as in [3],

(3) Here a n d t h r o u g h o u t the paper, M denotes a n y (in the sense of O(1)) positive c o n s t a n t which is i n d e p e n d e n t of E a n d of the n u m b e r of mesh points used when (1) is solved numerically. T h u s , M m a y have different values in different inequalities.

(2)

916 R. V U L A N O V I C A N D G. H O V H A N N I S Y A N It moreover holds (cf. [4]) t h a t

l u g ( x ) - u0(x)! _< M

+

x

x ,

(4)

where u0 is the unique C2(X)-solution of the reduced problem,

- b ( x , u)' + c(x, u) = 0, x C X, u(1) = 0. (5) Singularly p e r t u r b e d boundary-value problems arise in m a n y applications, see [5-7] for instance. P r o b l e m (1) has been used frequently as a model for testing different numerical methods for singular p e r t u r b a t i o n problems. In addition to the above-mentioned papers [3,4], some other more recent papers dealing with the numerical solution of (1) are [1,8,9]. We are interested here in one of K o p t e v a ' s results in [1], where the special case cu ~ 0 is considered. We represent this case by writing also c ( x , u ) = - f ( x ) . K o p t e v a ' s result is an a posteriori error estimate for the numerical solution of (1) with c(x, u) = - f ( x ) , obtained by the first-order upwind scheme. T h e error estimate is derived under the less restrictive smoothness assumptions b E C I ( X x R) and f E C 1 (X).

In Section 2, after introducing some further notation, we show t h a t K o p t e v a ' s approach can be applied to the general case c~ ~f 0 as well. However, we complete the derivation of the a posteriori error estimate differently, viz., we expand it and ignore all the terms of order higher t h a n one. We do this first in Section 3 for the special case c(x, u) = - f ( x ) and t h e n in Section 4 for the general case. In b o t h cases, we need smoother functions b and c (as indicated in our assumptions above) and we make use of the special discretization meshes of Bakhvalov or Shishkin types. T h e general case requires also t h a t the reduced solution u0 be taken into account. In Section 5, we present results of some numerical experiments. The results are discussed in Section 6 along with some other concluding remarks.

2. P R E L I M I N A R I E S

Let X g be a general discretization mesh with points xi, i = 0, 1 , . . . , N , where 0 = x0 < xl < . . . < x g = 1. Let a l s o X i = [Xi-l,Xi], i = 1 , 2 , . . . , N , h, = x i - x i _ l , i = 1 , 2 , . . . , N , hi = (hi + h i + l ) / 2 , i = 1 , 2 , . . . , N - 1, h g = h g / 2 , and h = m a x i h i .

We consider two special discretization meshes, b o t h dense in the b o u n d a r y layer. The first one belongs to the meshes of Bakhvalov type. It is generated by a suitable function A so t h a t xi = ,k(i/N), i = 0, 1 , . . . , N. A general description of mesh generating functions can be found in [8,10] for instance. For simplicity, we consider here, specifically,

{

A(t) = ~(t) .-- (q_--t), if 0 < t < c~, ~ b ( t ) : = T ' ( a ) ( t - a ) + ~ o ( a ) , i f a < t < 1,

el. [4,10]. Here, q is a mesh parameter, a fixed number in the interval (e, 1), and a is the unique number guaranteeing t h a t ~(1) = 1. Thus, A is a strictly increasing C 1 (X) function which maps X onto itself. Let X ~ denote the discretization mesh generated by the specified A.

The other mesh is of Shishkin type. Shishkin meshes are piecewise equidistant and therefore simpler, see [4,8] for instance. However, they produce somewhat less accurate results t h a n Bakhvalov meshes, cf. (7),(8) below. For problem (1), a Shishkin mesh consists of two equidistant parts, one fine over the interval [0, ~], and the other coarse over IT, 1]. Here, T is the transition point between the fine and the coarse parts of the mesh, ~- = (2e/13) In N . Then,

{2~

N ' 'i for i = 0, 1 , . . . , -~-, N r + (1-- ~-) 2i N N

, f o r / - 2 ' 2 + I , . . . , N , where we assume for simplicity t h a t N is even. Let this mesh be denoted by X g .

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A Posteriori E r r o r E s t i m a t e s 9 1 7

For b o t h types of meshes, h = h N <_ M / N .

By w N = {wiN}, we denote an a r b i t r a r y mesh function o n X N. For any mesh function, we assume t h a t Wo N = w N = 0. We discretize problem (1) using the s t a n d a r d upwind scheme, also known as the Engquist-Osher scheme [11],

E (D+wiN - D_wiN) 6 + b ( x i ' w N ) T N w N : = -h--i hi + c(xi,wiN) -- 0, i = 1 , 2 , . . . N - 1, (6) where and D + w N - 6+w N D_wiN -- 5 - w N hi+l ' hi '

+wiN

= W i + 1

N _ wiN,

_wiN = w, -wiN_,.

B y [8], the discrete problem (6) has a unique solution, which we denote by w N -- {wNi}. This solution is b o u n d e d uniformly with respect to & Let u N denote the piecewise linear interpolant of w N. Thus, u N C C ( X ) , it is a linear function on each interval X~ and u g ( x i ) = w N. for i = 0, 1 , . . . , N. If the special meshes are used, the following holds t r u e according to [9] (the same is proved in [8] but t h a t proof requires a smoother function b),

where L w N I ~,i - ~ ( x i ) [ _< M ~ , i - - - 0 , 1 . . . . , N , (7) 1, if X N = X N, L = In N, if X N ---- X N .

A n o t h e r property of the special meshes is

lug(x) - u~(x,-1)l _< M L , x e X,.

Analogously to the following form of the differential equation in (1),

- ( A u ) ' = 0, A ~ = ~ ' +

b(x, ~) +

c(t, ~(t)) dt,

the diseretization (6) can be written down as N N T N w i = A w i + 1 - A N w N f~i = O, i = 1 , 2 , . . . , N - 1, with (8) (9) (10)

(11)

(12)

2 ¢ ~ 2 v _ f . '

H

-<

~ := - E u " - 6(x, ~)' = f ( x ) , ~ e x , u(o) = ~(1) = 0.

T h e following result is crucial in her error analysis, N

A g w N = ¢ D _ w g + b ( x i , w N) + E h y c ( x j , w N ) , i = 1 , 2 , . . . , N . j = i

This form of the scheme is similar to the one in [9], which uses a more general definition of [zi. In [1], on the other hand, the operator A N is defined in a slightly more general way (for the case c ( x , u ) = - f ( x ) considered there). However, neither generalization is essential and we do not deal with t h e m here.

(4)

918 R. VULANOVIC AND G. HOVHANNISYAN

where

I1~11~ = e s s

sup

]u(x)l,

xEX

E s t i m a t e (12) immediately gives

N~II.= rain IIUIl~.

U:U'=u

2

for the general problem, where for

x C Xi, i = 1, 2 , . . . , N ,

(13)

~x 1

7 (X) : --~ (uN) / (X) -- b (x, 7z N (x)) -t- C -

c (t, u¢ (t)) dt

with an a r b i t r a r y constant C. Thus,

7 ' ( x ) = T u N (x)

- c ( x , u

N (x)) + c ( x , u ~ (x)).

Since from (13) it follows t h a t

2

II uN -- uellc~ <-- ~ ]1711oc ,

(14)

the

a posteriori

error estimate depends on how 7/is estimated.

We now t r a n s f o r m T/(X) for x E Xi analogously to [1]. First, we choose C as C =

A N u N ( x N )

SO that, according to (10),

ANuN(x~) = C

for all i = 1, 2 , . . . , N - 1. Then, we use the fact t h a t

(uN)'(x) = D _ u N ( x i )

for

x e Xi.

We get

j[x 1

7 ( x ) = - - z D - u N ( x ~ ) - - b ( x ,

uN (x)) + ANuN ( x i ) -

c ( t , u ~ ( t ) ) dt

N

1 = b (X,?~N (Xi)) _ b ( x , u N (x)) -I- E h j c ( x J ' u

N ( x j ) ) - ~x c ( t , u E (t)) dr.

j=i

Therefore,

where for

x C Xi

and

7 (x) = m (x) + 72 (x),

j~X xi

t

71 (X) : b

(t,

u N (t)) dt

(15)

N 1

72 (x) = E hjc (xj, u N (xj)) - ~ c (t, uE (t)) dr.

j=i

(16)

In Sections 3 and 4, we are going to use some a p p r o x i m a t e equalities ( - ) and inequalities ('<).

T h e y m e a n t h a t the terms we omit are negligible relative to

M h .

3 . T H E

C A S E

c ( x , u ) = - f ( x )

In this section, we consider problem (11). K o p t e v a ' s error e s t i m a t e in [1] is based on

II~/211~ -< (l[flI~ + IIf'll~) h

and

where

][r/1H~ _</3 m a x

15_uN (xi)] + B h ,

l<_i<N

(5)

A Posteriori E r r o r E s t i m a t e s 919

and

B =

max

I b~ (x, u) l.

xEX,lul<__M.

The above constant M. results from an

a priori

estimate of the numerical solution,

1 [2 ]]b(.,O)H~ + Ilft[oo],

II

-< M . :=

see [1]. Assumption (17) is not as serious a restriction as it may seem. It is introduced in this

form for simplicity since it is possible to find an a

priori

domain containing u~ and then the

boundedness of b~ is guaranteed for u in that domain.

Thus, assuming that

b E C I ( X

x I~) and f E C I ( X ) , and using (14), Nopteva proves the

following a

posteriori

error estimate:

HU N __ UeIIoo ~ ~2 [~ l_<i_<Nmax

15__U N (Xi) I -~- (B

+ Ilfl[~ + IIf'lloo) hi .

(18)

This estimate is valid on any mesh

X N.

We improve estimate (18) by expanding and approximating both zh and ~12. We do this under

the assumptions that the discretization mesh is either X N or X N and that b and f are smoother

functions. Our approximation of rh is given in the following lemma.

LEMMA 1.

Let b E C 2 ( X x ~) and let the discretization mesh be either X N or X f f . Then, for

x E Xi, it holds that

l

~1 (~) :- (x~

-

~) Yx

(~))

~ = ~ "

PROOF. Expand

b(t, uN(t)) '

in (15) about x~ to get

where

~]l(x):(xi--x) [J~b(x, uN(x))] +ri,

2 C : X i

The special mesh, (7), and (9) imply

L

I D - u N

(xdl ~ I D - [ug (x~)

_ u ~

( x d ] [ ÷

ID_u~

(xdl ~

Mh-- ~ ,

and therefore

Iril < M ( L / N ) 2

and this term can be ignored.

|

Lemma 2 deals with 72. This result is true not only on the special meshes but on all meshes

with

h <_ M / N .

C2(X) and

let

the

discretization mesh be

either X g or X N. Then,

for

LEMMA 2.

Let f E

x E Xi, it holds that

PROOF. Upon replacing

c(x, u)

in (16) with - f ( x ) , we get

where

1 N

72 (x) = t

f (t) dt - E h j f (xj) = ~l + ~2,

j = i

(6)

and @_ is the error of the trapezoidal formula for

f:~ f ( t ) dt. Therefore,

I @ [ _ < M N h

3 < M h 2.

Moreover, by expanding

f ( t ) in {1 around x~, it follows that

~2 - ~1 ± (xi - x) f (xi) - ~ f ( x i ) .

|

We can now prove the main result of this section.

THEOREM 1.

Let b E C 2 ( X × ~), f E C2(X), and let b satisfy the condition in (2). Let u~ be

the solution of (11) and let u N be the linear interpolant of the numerical solution of (6) on X N

or X s N and with c(x, u) = - f ( x ) . Then, the following approximate a posteriori error estimate

holds true:

IluN-u ll A ,<nh%xh max

I / ( x d l ,

[b(x, uN(x))']~=~+

. (19)

PROOF. Combining the results of Lemmas 1 and 2, we get

After maximizing the above right-hand side, we conclude t h a t

']

Then, the assertion follows from (14).

I

Numerical results in Section 5 confirm that the error estimate (19) is much sharper than

Kopteva's (18). Another advantage of (19) is t h a t it does not need the upper bounds for Ib=],

bu, and [fl. Note that the values of

[b(x,uN(x))']x==~ can be calculated easily after finding the

W N •

numerical solution { E,i}.

[b (x, u N (x))']~=x ~ = b~ (x~, u N (xi)) + b. (xi, u m (xi)) D - u g (xi)

= b~ (x i, wgi) + b~ (xi, wNi) D - w N i .

4 . T H E

G E N E R A L

C A S E

We now return to the fully quasilinear problem (1). In this case, r]2 cannot be treated in the

same way as in the previous section. Therefore, in this section, we make use of u0, the solution

of the reduced problem, and assume that ¢ << h, which is not a serious practical restriction. Of

course, the reduced solution may be used in other ways in numerical methods for problem (1),

see for instance [12,13]. We are interested here only in the numerical method given in (6) and in

seeing how the error of its solution can be estimated using u0. We assume t h a t u0 is known, but

even if it is not, its numerical approximation of at least second order can be used instead.

We first replace uE in (16) with u0. Because of (4), it follows t h a t

N 1

(x) "-- ~ hjc (xj, u N (xj)) - f

c (t, uo (t)) dr,

x e Xi.

~72

j = i J x

Then, the integral above can be modified and approximated like in the proof of L e m m a 2. This

gives

with

N

=

( x j , J

- e( j, 0 (x j ) ) } .

j = i

(7)

A Posteriori Error Estimates

921 THEOREM 2. Let b, c E C 2 ( X x R) and assume condition (2). L e t us and uo be the solutions of

(1) and (5), respectively. Also, let u N be the linear interpolant o f the numerical solution of (6)

on X g or X g . Then, if s << 1 / N , the following approximate a posteriori error estimate holds

true:

where

and

1 II uN

- ~ 1 1 ~ -

'~ ~ l_<,_<Nmax

m a x { A , , B , } ,

Ai = [hic(xi,uo (xO) + 2ai[

(20)

T h e estimate (20) can be modified. For this, we need the following auxiliary result.

LEMMA 3. L e t uo be the solution of the reduced problem (5) and let u N be the linear interpolant

o~ the numerical solution o~ (6) on X g or X ~ . T~en,

x E X i .

PROOF. Let u N denote the linear interpolant of {u~(x~)}. I t follows t h a t

f f ~

[u N (t) - uo (t)]

dt

_< M (I1 + 12 + I 3 ) ,

where

f

~'

dt

I1 =

[uN (t) - C (t)]

,

ffz ~

dt

I2 =

[ u y (t) - u~ (t)]

,

and

~

dt

5 =

[u~ (t) - uo (t)]

It holds t h a t Ij < M L / N 2, j = 1, 2. For I1, this follows from (7) and f o r / 2 , from (9). Finally,

13 <_ M ( s / N + s) because of (4).

|

TItEOREM 3. Let b,c C C 2 ( X × •) and assume the condition (2). Let uc and uo be the solution

of (i) and (5) respectively. Also, let u N be the linear interpolant of the numerical solution of (6)

o12 X N or X N. Then, if E << I / N , the following approximate a posteriori error estimate holds

true:

1 HuN--U~[[°°

~ fl l<i<Nmax

[2fl[(~_

[uN(xi)--u0(xi)]l ~-Ihic(xi,uo(2ci)) +2oil] .

(21)

PROOF. ,1, given in (15), can be rewritten as

. l ( x ) =

[ b ( t , u N ( t ) ) - b ( t , u o ( t ) ) ]

d t +

c ( t , u o ( t ) ) dt,

x E X i .

Then, it follows t h a t

(8)

922

with

and

R. VULANOVIC AND G. HOVHANNISYAN

~

3g i

01 (x) =

[b (t, u N (t)) - b (t, Uo (t))]' dt,

z • x ~ , (22)

j i x l

N

~x 1

02 (x) = ~2 (x) +

c (t,

uo

(t)) dt - E

h j c ( x j , u S ( x j ) ) -

c (t, uo (t)) dr,

x • X i .

j=i

Using N h < M and the trapezoidal formula again (cf.

following approximation of ~2,

hi c

f/2 (x) -" ai + -~- (xi, u0 (xi)),

x • Xi.

(23)

Let us now approximate r/1. Because of Lemma 3, for x • X~, it follows from (22) that

dt

= Jz

bu (t, "0, N (t)) [ D _ u N (xi) - U~o

(t)] dt.

We next replace U~o(t) with D _ u o ( x i ) creating a negligible second-order error,

f['

01 (x) "- [D_ (U N (Xi) -- U 0

(Xi))]

b u (t, U N (t)) dt,

x • X , .

From this, we get

t~ (x)r -< ~ I ~- [ ~ (x~) - uo (x~)] I,

x • X,,

(24)

where ~ is given in (17). We complete the proof using (24) and (23).

1 5. N U M E R I C A L

R E S U L T S

We consider three test problems, two linear ones and one nonlinear. The first linear problem

is of the type described in (11),

- E u " - u' = f ( x ) ,

The second linear problem is with c~ ~ 0,

- s u " - u ' + u = g ( x ) ,

Both problems have the solution,

the proof of Lemma 2), we get the

x • X ,

u ( 0 ) = u ( 1 ) - - - 0 .

(25)

x • X ,

u ( O ) = u ( 1 ) = O .

(26)

u~ (x) = ( e - 1)e-x/el

- e - l ~ e + e -1/~ + e~ '

which determines the functions f and g above. The nonlinear problem is a classical example due

to O'Malley [14],

(9)

A Posteriori Error E s t i m a t e s 923 T h i s p r o b l e m h a s b e e n u s e d i n m a n y n u m e r i c a l e x p e r i m e n t s , i n c l u d i n g [1]. I t s s o l u t i o n s a t i s f i e s u e ( x ) -~ U A ( X ) 4- O ( E ) , w h e r e

[(

)]

U A ( X ) = - - l n l + c o s - ~ - 1 - - e - x / ~ . I n a l l o u r n u m e r i c a l t e s t s , w e e v a l u a t e t h e e x a c t m a x i m u m e r r o r , _~ w N E = E ( N ) m a x I e , i - u ~ ( x i ) ] I < i < N - 1 w h e r e z~E = UA f o r t h e n o n l i n e a r p r o b l e m ( 2 7 ) a n d 5¢ = u e f o r t h e l i n e a r p r o b l e m s . W e c o m p a r e E t o t h e a poster~iori e r r o r e s t i m a t e s . I f E * d e n o t e s a n a p o s t e r i o r i e r r o r e s t i m a t e , t h e n w e c a l c u l a t e i t s e f f i c i e n c y a s E e x p e c t i n g t h a t E f t _ < 1. W e a l s o e v a l u a t e t h e n u m e r i c a l o r d e r o f c o n v e r g e n c e , [ E ( N ) ] O r d --- O r d ( N ) = l o g 2 [ E - - - ~ J a n d f i n d O r d f o r a l l a p o s t e r i o r i e r r o r e s t i m a t e s E * a s w e l l . A l l t h e r e s u l t s s h o w n h e r e a r e o b t a i n e d f o r E = 10 - 9 . D u e t o t h e z - u n i f o r m i t y o f t h e n u m e r i c a l m e t h o d s u s e d , t h e r e s u l t s f o r o t h e r s m a l l v a l u e s o f z a r e s i m i l a r .

Table 1. Errors, error e s t i m a t e s , t h e i r n u m e r i c a l orders of convergence, a n d error- e s t i m a t e efficiency values for (25) solved on X N w i t h q = 0.5.

N 32 64 128 256 512

E (19) Eft (18) Eft A Eft Eft of A*

O r d O r d O r d O r d

1.30E - 1 2.85E - 1 .46 9.99E - 1 .13 1.60E - 1 .81 .41

.94 .93 .99 .96

6.75E - 2 1.50E - 1 .45 5.04E - 1 .13 8.23E - 2 .82 .41

.98 .96 .99 .98

3.43E - 2 7.70E - 2 .45 2.54E - 1 .14 4.18E - 2 .82 .41

.99 .98 1.00 .99

1.73E - 2 3.90E - 2 .44 1.27E - 1 .14 2.11E - 2 .82 .41

.99 .99 1.00 .99

8.68E - 3 1.96E - 2 .44 6.36E - 2 .14 1.06E - 2 .82 .41 Table 2. Errors, error e s t i m a t e s , t h e i r n u m e r i c a l orders of convergence, a n d error- e s t i m a t e efficiency values for (25) solved on XB N w i t h q = 0.8.

N 32 64 128 256 512

O r d Ord Ord O r d

2.29E - 1 4.25E - 1 .54 2.43E + 0 .09 3.63E - 1 .63 .32

.97 1.00 .97 .89

1.16E - 1 2.12E - 1 .55 1.24E + 0 .09 1.96E - 1 .59 .30

.99 1.00 .98 .94

5.87E - 2 1.06E - 1 .55 6.29E - 1 .09 1.02E - i .57 .29

.99 1.00 .99 .97

2.95E - 2 5.31E - 2 .56 3.16E - 1 .09 5.21E - 2 .57 .28

1.00 1.00 .99 .99

(10)

924 R. VULANOVIC AND G. HOVHANNISYAN N 32 64 128 256 512 T a b l e 3. E r r o r s , e r r o r e s t i m a t e s , t h e i r n u m e r i c a l o r d e r s o f c o n v e r g e n c e , a n d e r r o r - e s t i m a t e efficiency v a l u e s for (25) s o l v e d o n X N . E O r d 1 . 7 1 E - 1 .75 1 . 0 2 E - 1 .79

(19)

E f t O r d 9 . 8 8 E - 1 .51 6 . 9 2 E - 1 .63 5 . 8 9 E - 2 4 . 4 7 E - 1 .82 .72 3 . 3 3 E - 2 2 . 7 2 E - 1 .85 .77 1 . 8 5 E - 2 1 . 5 9 E - 1

(18)

E f t A E f t E f t o f A* .35 .17 .29 .15 .26 .13 .24 .12 .23 .12 O r d .17 1 . 6 7 E + 0 .70 .15 1 . 0 3 E + 0 .74 .13 6 . 1 7 E - 1 .79 .12 3 . 5 7 E - 1 .82 .12 2 . 0 2 E - 1 O r d .10 4 . 9 4 E - 1 .51 .10 3 . 4 6 E - 1 .63 .10 2 . 2 3 E - 1 .71 .09 1 . 3 6 E - 1 .77 .O9 7 . 9 6 E - 2

We use the problem (25) to compare our estimate (19) to K o p t e v a ' s estimate (18). All the quantities needed for (18) are easy to find. T h e comparison is given in Tables 1-3 on different discretization meshes. In her paper [1], Kopteva does not calculate (18), b u t uses instead the quantity,

A = m a x ] 6 _ w 5 ] , l < i < N - 1

although there is no theoretical guarantee t h a t A is an upper b o u n d of the error. We, too, include A in all our tables, as well as

A.

= 7

2D

A,

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since it seems reasonable to preserve the coefficient multiplying A in (18). Therefore, A* is a theoretically safer error estimate t h a n A, but not as safe as our estimates. In the linear test- problems, A* = 2A.

C o m p a r i n g Tables

1-3,

we can see t h a t the Bakhvalov-type mesh XB N produces much better results t h a n the Shishkin mesh X s N. On XB N, the error estimate (19) has greater efficiency for q = 0.8 t h a n for q = 0.5. Greater values of the parameter q cause greater density of the mesh in the b o u n d a r y layer. In the remaining tables, we use only XB N with q = 0.8.

K o p t e v a ' s estimate (18) cannot be applied to (26) and (27). We use these problems to compare our two estimates (20) and (21) to A and A*. The results are given in Tables 4 and 5.

For the nonlinear problem (27), the quantities ~ and ~ are found as follows. ~ = 1 because the function identically equal to 0 is a lower solution of (27). In [4], t3 is estimated as p = exp(-~r/2), but this makes the coefficient 1 / ~ in (20) and (21) too large. Therefore, we t r y to estimate ~ more precisely. It is easy to verify t h a t the reduced solution of (27),

(

u o ( x ) = - l n 1 + c o s T a b l e 4. E r r o r s , e r r o r e s t i m a t e s , t h e i r n u m e r i c a l o r d e r s o f c o n v e r g e n c e , a n d e r r o r - e s t i m a t e efficiency v a l u e s for (26) s o l v e d o n X N w i t h q = 0.8. N 32 64 128 256 512 E (20) E f t (21) E f t A O r d O r d 1 . 7 3 E - 1 4 . 2 5 E - 1 .95 1.00 8 . 9 1 E - 2 2 . 1 2 E - 1 .98 1.00 4 . 5 2 E - 2 1.06E - 1 .99 1.00 2 . 2 8 E - 2 .99 1 . 1 4 E - 2 5 . 3 1 E - 2 1.00 2 . 6 5 E - 2 O r d .41 4 . 7 6 E - 1 1.07 .42 2 . 2 7 E - 1 1.05 .43 1 . 1 0 E - 1 1.02 .43 5 . 4 1 E - 2 1.01 .43 2 . 6 8 E - 2 O r d .36 3 . 6 7 E - 1 .90 .39 1 . 9 7 E - 1 .95 .41 1 . 0 2 E - 1 .97 .42 5 . 2 1 E - 2 .99 .43 2 . 6 3 E - 2 E f t E f t o f A* .47 .24 .45 .23 .44 .22 .44 .22 .43 .22

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A Posteriori Error Estimates

Table 5. Errors, error estimates, their numerical orders of convergence, and error- estimate efficiency values for (27) solved on X g with q = 0.8.

925 N 32 64 128 256 512

Ord Ord Ord Ord

1.26E - 1 4.91E - 1 .26 6.36E- 1 .20 1.81E - 1 .69 .17

.94 1.00 1.13 .80

6.53E - 2 2.45E - 1 .27 2.91E - 1 .22 1.04E - 1 .63 .16

.97 1.00 1.10 .89

3.34E - 2 1.23E - 1 .27 1.36E - 1 .25 5.63E - 2 .59 .15

.98 1.00 1.06 .94

1.69E - 2 6.15E - 1 .27 6.49E - 2 .26 2.93E - 2 .58 .14

.99 1.00 1.04 .97

8.50E - 3 3.07E - 2 .28 3.16E - 2 .27 1.50E - 2 .57 .14

is at the same time a lower solution of the problem. Because of this a n d uo(x) > u0(0), x ~ X , we can take ~ -- 1/2, so t h a t A* _-- 4A for the n o n l i n e r test-problem.

6. C O N C L U S I O N

T h e n u m e r i c a l results of the previous section confirm t h a t we have created a r o b u s t a posteriori

error e s t i m a t e for problems of type (1) w h e n t h e y are solved n u m e r i c a l l y using the u p w i n d discretization (6). Tables 1-3 clearly show t h a t our e s t i m a t e for the case c~ - 0 is superior to K o p t e v a ' s estimate (18). It is n o t surprising t h a t t h e q u a n t i t y A, which K o p t e v a uses in numerical experiments in

[1]

i n s t e a d of her error estimate, has t h e best efficiency. However, A is n o t a theoretically safe estimate. T h e reliability of o u r e s t i m a t e s is t h e o r e t i c a l l y well-founded, and, moreover, our estimates get very close to A in some cases (see Tables 2 a n d 4). A does not even c o n t a i n the coefficient 2~/f~ t h a t is present in (18). We i n c l u d e this coefficient in the q u a n t i t y A* defined in (28). In all tables, our estimates are either as efficient as A* or superior to it. Tables 4 a n d 5 show also t h a t our estimates (20) a n d (21) are relatively close, p a r t i c u l a r l y for larger values of N . E s t i m a t e (21) is somewhat worse, as should be expected.

We achieve the i m p r o v e m e n t over K o p t e v a ' s result in [1] by using special discretization meshes of either Bakhvalov or Shishkin type. I n the case w h e n c ( x , u ) = - f ( x ) , we e x p a n d the com- p o n e n t s 71 a n d 72 (see (15) a n d (16)) of a n initial theoretical error e s t i m a t e a n d keep only O ( h ) - t e r m s . O u r estimate is at the same t i m e easier to calculate a n d is therefore more practical. Unlike (18), it does n o t require the u p p e r b o u n d s for Ib~t, bu, a n d Ifl. Moreover, we generalize K o p t e v a ' s approach, since no e s t i m a t e is derived in [1] for t h e case c~ ~ 0. We accomplish this using the solution of the reduced problem (5) a n d the a s s u m p t i o n t h a t s << 1 / N .

e-uniform numerical methods, similar to those considered in this paper, have b e e n used also for s i n g u l a r l y p e r t u r b e d convection-diffusion problems in two space d i m e n s i o n s (see [7,15-18] for instance), as well as for systems of s i n g u l a r l y p e r t u r b e d o n e - d i m e n s i o n a l convection-diffusion equations such as the system considered in [19]. References [7,15] are general references which deal m a i n l y with finite-difference methods. F i n i t e differences are used also in [16], whereas [17,18] are examples of m e t h o d s based on finite elements a n d finite volumes, respectively. Ohlberger [18] deals with a n o n l i n e a r problem in time a n d two or three space d i m e n s i o n s a n d even proposes an a posteriori error estimate. However, the efficiency of this e s t i m a t e is relatively low. It is therefore a n a t u r a l question whether the a posteriori error-estimate t e c h n i q u e presented here can be applied to these more general problems. This is a possible topic for f u t u r e investigations. T h e m e t h o d we used to improve K o p t e v a ' s e s t i m a t e is relatively e l e m e n t a r y a n d c a n certainly be e x t e n d e d to more dimensions or systems of differential equations; t h e key issue is to o b t a i n s t a b i l i t y inequalities like (12) or (13). A different approach c a n be found i n [16], where K o p t e v a provides a detailed error-analysis for linear t w o - d i m e n s i o n a l convection-diffusion problems. She derives a n error-expansion for the u p w i n d finite-difference scheme on a S h i s h k i n - t y p e mesh. T h e

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main purpose of this result is to justify the Richardson extrapolation of the numerical solution, but it can be used also for an error estimate. This estimate is perhaps not so practical since two reduced problems have to be solved. Another question is whether this approach can be applied to nonlinear problems. There exists also an empirical error-estimate method, described in [7,20], which can be used for a wide range of singular perturbation problems.

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