A finite element method with a large mesh-width for a stiff two-point boundary value problem

A finite element method with a large mesh-width for a stiff

two-point boundary value problem

Citation for published version (APA):

Groen, de, P. P. N. (1978). A finite element method with a large mesh-width for a stiff two-point boundary value problem. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 7801). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1978

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EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics

Memorandum 1978-01 April 1978

Preprint

A finite element method with_ a large mesh-width for a stiff two-point boundary value problem

by

P.P.N. de Groen

Eindhoven University of Technology Department of Mathematics

PO Box 513, Eindhoven The Netherlands

I /

boundary value problem.

Abstract:

We describe a finite element method for computation of numerical approximations of the solution of the second order singularly perturbed

two-point boundary value problem on I,lJ

EU" + put + qu

=

f, u(-I)

=

u(I)

=

0,

o

< E « 1, (' = d

I

dx) • On a quasi-uniform mesh we construct exponentially fitted trial spaces which consist of piece-wise polynomials and of exponentials which fit

locally to the singular solution of the equation or its adjoint. We

discretise the Galerkin form for the boundary problem using such exponentially fitted trial spaces. We derive rigorous bounds for the error of discretisation with respect to the energy norm and we obtain superconvergence at the

mesh-points, the error depending on £, the mesh-width and the degree of

the piece-wise polynomials.

Subject Classifications AMS(MOS): 65LI0, 34L15.

Running head (short title): Galerkin methods and exponential fitting for singular perturbations.

I. INTRODUCTION.

A finite element method using piecewise polynomials as trial functions is in general not very adequate for application to "stiff" problems, i.e.

to differential equations in which the coefficient of a principal part 1S small

~n comparison with the mesh-width and the coefficients of minor terms. The reason

is that such smooth functions as polynomials are, cannot approximate

boundary layers and other singular behaviour of the solutions very well if the mesh-width is kept large in comparison with the inverse of the slope. For better approximations Hemker [5J devised so-called exponentially fitted methods in which the solution space and the test space contain local

approximations to the singular part of the solution. and of the Green's function respectively. He performed numerous numerical experiments with such types of elements and obtained good results. In this paper we shall prove the asymptotic validity of such methods for linear singularly perturbed two-point boundary value problems (without turning points). We shall restrict ourselves to equations of second order, but the proofs can easily be generalized to problems of higher (even) order.

lao THE PROBLEM. On the real interval

[-I,IJ

we study the singularly

perturbed two-point boundary value problem (1. I) L u := eu" + pu' + qu == f,

e:

u(-l)

=

u(l)

=

0, (' = d/dx) ,

where £ is a small positive parameter and the coefficients p and q are

smooth functions. Although our analysis and proofs are valid if p and q

are smooth functions of x and E and if either p does not have zero's

in

[-I,IJ

or q - ~p' is positive, we shall assume for simplicity that p and q

CIO

are C -functions of x alone which satisfy

(1.2) p(x) ~ PO > 0 and q (x) - ip I (x) ~

Y

X E

[-I,IJ.

If P does not have a zero,the first condition can be obtained by inversion of the interval and the second condition can be obtained by

the transformation u(x)

=

vex) eYx ; if Y is sufficiently large and if £

is small enough, the new equation satisfies (1.2). (We remark that such a transformation is performed only for the ease of proving and that it is not necessary and even may be bad in actual computations).

If P ~anishes somewhere in [-I,IJ, the problem is of turning point

type. It has for each £ >

°

a real spectrum, bounded below by the minimum

of q - ~p', which does not vanish at infinity in the limit for £ + +0,

cf. [3J or

[4J.

In this case the analysis we shall give is true only if

all eigenvalues are positive, i.e. if q - !p' > 0.

It is well-known, cf. [2J, [3J or [6J, that condition (1.2) implies

existence of a unique solution U of problem (1.1) for each £ > 0 and each

£

square integrable f. This solution U converges for £ + +0 to the solution

£

of the reduced problem

(1.3) pu' + qu

=

f, u(-I)

=

0

uniformly on the subdomain [-1, aJ for each a < 1 and it displays a boundary

layer at the right-hand end of the interval. Moreover, U£ is a solution

O.

J) if and only i f it is the solution of the Galerkin (orwea~) form

(1.4) B (u,v) := £(u',v') + (pu' + qu,v)

=

(f,v)

£

wher~

(.,-) denotes the

~sual

inner product in L2(-I,I).

. I

Vv E HO(-I,I),

n

lb. TRIAL SPACES. Let 6:= {xi}i=O be a partition of the interval [-l,lJ,

-I = Xo < x. < x. I < x = I

1. 1.+ n ( i

=

J, ••• , n-J ) ,

with meshwidth h := max IXi - Xi-II and such that

i

min

i

of

Let u

1i denote the restriction of u E L

2

(-1,1) to the subinterval [xi_1,xiJ.

We define P

k as the set of polynomials of degree not greater than k,

For the partition A we define the set of continuous piece-wise polynomials

h P

k by

( 1 • 5) P: :... {u € HI (- 1 , J)

I

u liE P k' i

=

1, 2 , ••• , n} •

In connection with problem (1.1) and the partition 6 we define the exponential

functions

w:

and

w~

(i

=

O,I, ••• ,n) by

1. 1.

± exp{± p(x. )(x-x.)!e:} - exp{± p(x. )(x.:;: J - x.) Id

( 1. 6) w. (x) : = _ _ _ _ _ 1 _ _ ...;;1;;...-. _ _ _ _ _ ...;1.;;;...-..;;1~_...;1.;..._

1. ..

The exponential

w:

is the solution of

1.

- EU" + p(x.)u' = 0, u(x.)

=

I,

1. 1 u(x. 1) 1-

=

0;

it is the first term of a local boundary layer expansion for L u

=

0

g

and it decays to the left. The exponential w. is the solution of

1

- EU" - p(x.)u' = 0,

1. u(x.) 1 = 1,

it is the first term of a local boundary layer expansion for the adjoint

* . .

problem L u = 0 and 1t decays to the rlght.

e:

Adding these functions to

P~

we obtain the exponentially fitted trial spaces

h h Ek and F_k, cf. [5, ch 3.4J, (1.7a) (1. 7b) Eh k Fh k := {u E ;= {u E 1 HO (-1,1) 1 HO (-1,1) ul i E span ul i E span (P_{k '} w.) , + ₁ i = 1 , ••• , n} , (P k, w -:-1.-1)' i

=

l, •.. ,n};

their elements satisfy the boundary conditions of problem (1.1).

If g

<

h, the boundary layer in the solution of (1.1) is contained in the interval [x I,IJ almost entirely. In such a case the exponentials

n-in the reman-inn-ing n-intervals hardly give any contribution to the approximation and can be omitted. Therefore we define the partially fitted trial space

Eh by

k,p

(1.7c)

E~,p:=

{UE

E~

I

ul

i E Pk, i .. I, •••

,n-D.

In order to have test spaces whose dimensions match to the dimension of

h Ek ,we define ,p (1.7d) Fh := {u E Fh

I

ul i E span (Pk-1 ,w~), i = 1 , ••• , n-1 } , k-I,p k (1.7e) ph k,p := {u E

p~+llul

i E Pk, i .. 1 , ••• ,n-l } ;

in these spaces the degree of the polynomials on [xn_l,lJ is enlarged

by one (with -respect to

F~_I

and

P~

respectively).

We shall not go into the question, how to choose optimal bases in these trial spaces and how to evaluate the inner products in the _bilinear

form B (.,.) _g in actual computing. These questions are settled satisfactorily

in [5J by Hemker.

h h

Ie.

THE RESULTS. With a given solution space S and test space

V

of equal

finite dimension the discretized form of (1.4) is to find u E Sh such that

Existence and uniqueness of a solution of (l.S) is guaranteed by an a priori estimate of the following type:

(l.9) VUE Sh 3 v E Vh such that B (u,v) ;:z: C lIull II vii

E E £ with C > 0,

where \I ull2 :- £II ut_ll2 _{+ II u112}

is the energy norm associated with B • Such an

£ h £ h

estimate is an immediate consequence of assumption (1.2) if S = V • We

shall prove its validity also in several cases where Sh

f

v

h under the

assumption that h + e:/h is small enough.

If Sh

=

Vh

=

E~,

i.e. if both trial spaces are fitted to the singular

solution of L u = 0, we

£ obtain the error estimate in the energy norm

(1.10) lIuh - u II

E E E

if £

<

h the choices

1. e. partial fitting of the trial spaces, yield the same result.

If Sh

=

Vh

=

F~,

i.e. if both trial spaces are fitted to the singular solution

of the adjoint equation L*u

=

0, we obtain an error estimate at the mesh

E points:

(1.11) luh(x.) - u (x.)1 S C(E + hk) , i

=

1,2, ••• ,n-l •

E ~ E ~

I f both ways of fitting are combined, i.e. i f

v

h ;::)

F~

and Sh :,)

~.~

or

Sh :,) Ekh ,we obtain the error estimate (1.10) in the energy norm and

super-,p

convergence at the mesh points: (1.12)

The main point of the proof of (1.]0) is to show that the solution

space contains a good approximation of the exact solution U • Using the lower E

bound (1.9 ) and a suitable upper bound for B we show that this good

£

approximation differs only little from the Galerkin approximation Uh• In the

. E

proofs of (1.11) and (1.12) we use the trick by which Douglas

&

Dupont prove

their superconvergence result, cf. [lJ. This trick hinges on the fact that

the Green's function of problem (1.]) can be approximated accurately (as

a function of ~) by an element of the test space if x is located at a mesh

point. Since G (x,,) (with x fixed) is a solution of the adjoint problem

£

(1.13) L G (x,,)

*

=

0 , G (x,-l)

=

G (x,l)

=

0,

E £ X £ E

where a is Dirac's a-function at x, this trick can be employed succesfully

x

in this singular perturbation problem only i f the exponentl.als' in the test

NOTATIONS:

C denotes a generic (positive) constant, which may be different at each occurrence.

2 .

L (a,b) denotes the set of square integrable functions on the interval (a,b),

equipped with the usual inner product (.,.) and norm lI·n , b

(I. 14) (u,v) :=

J

u(x) v(x)dx, II ull := (u,u) . ~ •

a

If it is not mentioned explicitly otherwise, we assume a

=

-1, and b

=

I.

Hk(a,b) is the set of functions in L2(a,b), whose k-th derivative is square

integrable. In HI(-I,I) we use the €-dependent inner product (o,o)€ and norm II oil

€

(1. 15) (u,v) := E(U' ,v') + (u,v), II ull := (u,u)! •

E £ E

The restriction of this inner product and norm to the subinterval

[x. I ,x. ] of the partition ~ is denoted by (.,0) . and non· . :

1- 1 £,1 £,1

(1.16)

x.

(u,v) .:=

f1

E,1

2. EXAMPLES.

In order to gain some insight in the features of the trial spaces ,defined in section Ib, we shall study their use in the simple problem,

cf. [5, example 3.4.3],

(2. I) e:u" + u' :: 1, u(-l) ~ u(l)

=

0,

in which we know the exact solution U ,

. e:

(2.2) U (x)

:=

x - (2 e x/e: - e lIe: - e -1/e:)/( e lIe: - e -lIe:) •

e:

In all examples we shall use the partition 8:- {-1,0,1} and we shall keep

the dimension of the trial spaces very low.

2a. The trial space Sh

=

v

h ::

E~

=

span {Xe:} ,

(2.3)

x

(x) :=

e:

x/e: -lIe:

e - e if -1 ~ x ~

°

I - e(x-I)/e: if

°

~ x ~

yields the "approximation" (2.4)

which happens to be exact at x :: 0. For the rest the approximation,is

rather poor.

2b. Approximation becomes better, if we enlarge the degree of the polynomials.

h h h }

The trial space S :: V

=

El :: span {X , _e: ~ _e: " _e: ,X as above, _e:

r

i f -1 ~ x ~ 0,

(2.5) ~ e: (x) :=

-1 e: -lIe: (x-I) Ie:

x(J-e I ) + e - e if

°

~ _x ~ I ,

{ :X

+ 1)(1 - e -1/£) + e -lIe: - e 'x/e: i f -I ~ x ~ 0,

(2.6)

'e:

_(x) :=

if

°

~ x ~ I,

contains the exact solution, such that the discretized problem yields the exact solution in this example.

2c. Partial fitting, however, yields also a good result in this case.

·The trial space Sh

=

Vh

=

Eh

=

span {I - lxi, f;; }, f;; as above, yields the

l,p £ £

approximation (2.7)

a := ( 1 _ 3 ₁ + ₊ 2£ -1/£)/ _{2£ e y,}

which is very good if £ is small.

S := 2/y, y := I + - 2£ _{+ 2£ e}

-1/£

_,

2d. Fitting of the exponentials in the trial space to Green's function, i.e.

h h · Sh h h { } t e c Olce

= V

=

FO

=

span ~£' (2.8) :=

{I -

e-(; + I)h

-;/£

-1/£

e - e i f -) ~ , ~ 0, if

o

~

;

~ I,

yields the "approximation"

which approximates U very badly, except at x := 0, where it is exactly

£

equal to U • This is due to the fact that ~ (.) is equal to (a constant

E: e;

multiple of) Green's function G(O,') (i.e. at x

=

0) of problem (2.1). This

example illuminates that fitting of the trial space to Green's function can yield an approximation which is good at the mesh points but possibly very poor in other points (especially in the boundary layer region).

2e. Fitting to Green's function in conjunction with partial fitting to the

h h h h

solution, with the choice S = V

=

EO + FO

=

span{~ ,

W

},~ as above,

,p £ E: €

0 i f -I ~ x ~ 0,

(2.9) tJi

(x)

:=

E:

1 + e

-1/£

- e (x -

1)/£

- e

-x/£

i f

o

~ x ~ _{1 ,}

yields the approximation

(2.10)

u

h

=

aw +

~

/(1 + e-1/E) ,

e: £ £

1 -

2~

+ (I

+.2~)e-I/E:

I

11

which is better than the examples 2a and 2d. As in these cases the approximation

3. A PRIORI ESTIMATES.

In the proofs of the error estimates for approximate solutions of

problem (1.1) we employ normrinequalities which display the relation between

the operator L , the bilinear form B and the "energy norm" (or "natural

€ €

norm") 11·11 • This norm is defined by

€

(3. 1) II ul 2 2 2 _£ : = £11 u'li _. + II ul

and it is related to the usual norm in H1(a,b) as follows:

(3.2) £

i

II ul I s II ul £ s II ul

a

+ £ ill ul 1 '

. 1

V U € H (a, b).

LEMMA I: Every u E H2(a,b) (with b > a) satisfies the inequalities

(3.3) II ul 2

£

s

(C{

C{

PROOF: The functional u 1+ u(a) is continuous in HI (a,b) and it satisfies

Sobolev's inequality b

(3.4) lu(a)12

:=

f

:x

{~::

lu(x)12}dx s 21lulllu'li + lIuI 2/(b - a).

a

2

Hence, any u E H (a,b) satisfies at x

=

a the inequality

(3.5) _I u' (a) ₁2 s

£'

2 II u'li (II pu'li + II qui + II L£ ul) + II u'li . / (b - a). 2

and at x

=

b it satisfies the same inequality. Integrating (L Utu) by parts,

€

(3.6) (L u,u) = £lIu'li 2 + «q - !p')u,u) + [Opu - €U')u] b ,

€ a

and using the inequality

we find from (3.4), (3.5) and (1.2) a constant C >

a

such that (3.3) is

2

LEMMA 2:

(3.7)

(3.8)

(3.9)

1

For all u,v E

H

O

(-I,I)

the bilinear form

Be:

satisfies the estimates

B (u,u)

~

lIuU2 e: e: B (u,u)

~

ClluU2 e:

e:

B (u,v) ~ e: CII u1I₁11 Y.:1I e:

PROOF: The lower estimate (3.7) and the upper estimate (3.8) follow from

formula (3.6) and assumption (1.2). The weaker upper bounds (3.9) follow

from the definition (1.4), q.e.d.

This lemma does not give a lower bound for the restriction of B to

h h e:

Ek

x F_k, In order to obtain such a lower bound, we have to use the property

that each element of these spaces locally can be written as 'the sum of a polynomial and an exponential part, which are approximately orthogonal with respect to the local inner product (.,.) . on [x. l'x.], cf. (1.16).

e:,l 1.- 1.

(3.10)

ul' ...

_{IT. + (l·W.} + _{+ S·w.} - _{l '}

1 1. 1. 1. 1. 1.- (i III 1,2, ••• ,n)

with IT. E pk and w± as in (1.6), the parts satisfy the estimate

1.

(3. 11) _{lilT ·11} 2 _{• + II} (l • w + ./1 2 • + /I 13 • w • - 111 2 • ~ II Ull 2 , . . - ; ; -• (l + C y dh) ,

1 E~1. 1 1. E,1 1. 1- e:,1 e:,l

provided 0 < £ ~ h ~ 1.

PROQF: Since IT. is a polynomial it satisfies the estimates

1

(3.12) _{hllTl·(x)1 + hllTl!(x)}

I

~

_{C h! IIlT.1I J •}

~

_{ClIlT.1I . min (It/h, {h/e:),}

1 : , l 1 E,l

for all x E [xi-1,x

i], provided e: ~ h; this implies

(3.13)

I

(IT . , ( l .

w:

+ 13.

w: ) .

I

~

c

1£

/h II IT .11 , (

I

(l l'

I

+ 113 1,

I,) .

The exponentials satisfy the estimates + 2 IIw·1I . ]; £,1 - 2 II w·1I _{1: £,1} '+1 = !<p(x.) + e:/p(x.»(1 - exp{±2p(x.)(x''''I-x.)/d) ;:: C > 0, 1 1 1 1T 1 (3.14) (3.15)

(w.,

+ w. - I) .:S; C h exp {p(x.)(x. 1 - x.)/d. 1 1- £,1 1 1- 1 +

These estimates imply that the cosines of the angles between

n.,

w. and

1 1

W

i-1 are of the order O(/e:/h) for £/h + O. Moreover, if e: :s;h, those angles

are bounded away from zero, as can be seen by computing the components of

+

-w. and w., which are orthogonal to P

k and to each other with respect to the

1 1

inner product (.,.) . on [x. l,x.J. This proves the estimate (3.11), q.e.d.

e:,1 1- 1

h h h

We now define a mapping T from Ek onto Fk as follows. If the restriction

·h of u E Ek to [x. l'x.] is written as 1- 1 + ul'

=

n. + a. w. 1 1 1 1 h

where n. is a polynomial, then we define the restriction of T u by

1

(3.16) (Thu)

I' :

= 1T. + a. 1 1 1

where P

k is the k-th Legendre polynomial and ~.(x) 1 := (2x - x. -1 h x. 1-I)/(x. -h 1 x. 1)'

1-with the aid of this mapping we obtain a lower bound for Be: on Ek x F

k:

LEMMA 4: A constant y > 0 exists such that

(3. 17)

provided 0 < h + £/h :s; y.

PROOF: Since the Legendre polynomial satisfies the identity 1

f

(P

k

(t»2dt

=

k(k+l),

-1

its norm satisfies the estimate

(3.18) IIPk(~·(·»1I 2 . 1 e:,1 x. - x. 1 2Ek(k +1) = _--'-_ _ -.':.. + 1 1 - :s; x. - x. 1 1 1- 2k+ 1

In conjunction with the previous lemma this implies (3.19) II Thull 2

~

II ul1 2 (1 + Ch + Ce:/h).

e: e:

Expanding the polynomial TI. in (3.16) in the Legendre polynomials we easily

1 .

see that Th is invertible and that its inverse has the same form as Th has; this implies the lower estimate

(3.20) II T h 2 ull ~ II ull 2 (1 + Ch + Ce: /h) -I •

e: e:

Defining the function~. by

1 (3.21) ~.(x) := 1 0, if x

f

[x·1,x.] 1- 1. i f x E [x. ) ,x.], 1. - 1.

we find the identity

(3.22) B (u,T u) h

=

B (u,u) + £ e: n

L

i=1 a. B (u,

~

.)

~

"ttli 2 -1 e: 1 e: n

I

i=1

la.B

(u,~·)I. 1. £ 1.

Inserting in this inequality the estimate (3.23)

which will be proved below, and using the estimates (3.11) and (3.20) we

see that (3.17) is true if h + £/h is small enough.

It remains to prove formula (3.23). Because the support of ~. is

1.

contained in [x. l'x,], we can integrate B by parts once,

1- 1. e:

(3.24) B (u,~.)

=

(L u,~.),

£ 1 e: 1

and we can estimate the parts of (L u,~.). Since TI. is a polynomial of

e: 1. 1. 2

degree at most k, its derivatives are orthogonal to Pk(~i('» (in L -sense).

Hence, by analogy to (3.12-13) we find

(3.25a)

₁

_{(E: TIi>~i)}

_I

=

I

dTI'~, W. + + _{(-1) wi - 1}k -

)1

~ _C

h

E: _{II TI .11 . ,}

1. 1 1. £,1

(3.25b)

I

(p TI!,~·)I _{1 1} = _{I «p - p(x.)}TI!, Pk(f,;i» - (TI!, 1} w. + + {-l) k -Wi-I)

I

~

1. 1 1.

~ _{C h}

~I

_{TI .11 0 . + C (E: /h)} ~ _{II TI .11 . ,}

(3.25c) l(q TT.,1/I.)I $; C

h~

IITT.II

O • + C(e/h)i IIn.11 ••

1 1 1 ,1 1 e , l

The exponential

00:

satisfies the estimate

1

(3.25d) (L .

w.,

+ 1/1 .) $; Ce. e l l

Inserting these estimates in (3.24) we find inequality (3.23), q.e.d.

Likewise we can derive lower bounds for B on Ekh x ph and on

e ,p k,p

Eh _k,p x Fh _{k-I,p' we} t:t d _{ecompose t e restrlc lons} h . t ' f Fh and u E Eh l'n

0 v E k-l,p k,p

a polynomial plus an exponential as before,

+

ul _n

=

n _n + a _{n n} w , v

I'

= X. + 13. w . 1 (i = 1, ••• , n-J ) , ~ 1 1

1-and we define the mappings Mh

Eh by

h h h h

from Ek,p onto Pk,p and N from Fk-1,p onto k,p

Uli' if i=l, .•• ,n-I,

(3.26a) (M h u)li :=

if i=n,

X. +

i

13 .(_I)k(P

k(E.:·(on - Pk 1(E.:.(·»), i f i=l, ••• ,n-J,

1 1 1 - 1

(3.26b) (N h V)li :=

LEMMA 5: A constant y > 0 exists such that

(3.27a) (3.27b)

Be (u,Mhu)

~

ill

ul ell

~ul

e

Be(U,(Nh)-lu)

~

iUul ell (Nh)-lul e

provided 0 < h + e/h S y •

h

V U E Ek _,p '

i f i=n.

PROOF: The proof follows the same lines as the preceeding proof. Expanding the polynomials in (3.16) in Legendre polynomials it is easily seen that Mh

and Nh are invertible. Using (3.18) and lemma 3 we easily find the estimates

(3.28a). II

~UI;

$;" UI; (I + Ch + Ce/h)

(3.28b) II Nhvll; $;11 vii

~

(1 + Ch + Ce/h)

VUE Eh

k,p

h

We remark that the norms of the inverse operators are of the order

O«h + E/h)-!) due to the difference between the orders of (3.14) and (3.18).

By analogy to (3.22) we have

h h

B (u,M u)

=

B (u,u) + B (u,M u - u).

E E €

Since Mhu - u has the support [x 1,IJ and

n- since ~ n in (3.26a) is of degree

k at most, the analogues of (3.25) apply. In conjunction with lemma 3 and estimate (3.28a) this implies (3.27a).

In order to prove (3.27b) we set v:= (Nh)-lu. By analogy to (3.22) we now have

h -) h h

B (u,(N) u) = B (N v,v)

=

B (v,v) - B (v - N v,v).

€ € E €

S · 1nce v - Nhv 1S zero at the mesh-po1nts an 1S smoot ot erw1se, we can . . d • h h • integrate by parts,

and we can write the right-hand side as the sum of restrictions to the subintervals, which can be estimated in the same way as in (3.25). In conjunction with lemma 3 and estimate (3.28b) this implies (3.27b), q.e.d.

4. BEST APPROXIMATIONS IN THE TRIAL SPACES AND ASYMPTOTIC EXPANSIONS.

In this section we construct asymptotic approximations to tpe solution of problem (1.1) and to its Green's function. From these asymptotic approxima-tions we derive error bounds for the best approximaapproxima-tions in the trial spaces

Ekh and Fhk of U _e:: and G • In the construction we use the method of "Matched _e::.

Aymptotic Expansions". Since this method is well-known, cf.[2,3 & 6], we

shall not give a detailed explanation of it.

The asymptotic approximation of U consists of a regular and a boundary

e:

layer expansion,

(4.1)

where p:= (x - I)/e:: is the boundary layer variable. Substitution of the

regular expansion in the equation yields the system of equations

(4.2)

p rj + q rj

=

rj_I' rj(-l)

=

0 , j

=

1,2, ••••

If f is sufficiently smooth, we can solve the system recursively, finding

x x

rO(x) ..

J

f(t) K(x,t)dt,

-I

rj(x)

=

f

rj_l(t) K(x,t}dt,

-1

where K is the kernel

x K(x,t) :=

p(l~)

exp{

_{- J}

If R . is the solution of e:,J (4.3) L u .. f, e:: u(-l) .. 0, t

it satisfies by (3.3) the estimate

(4.4)

t

i-O e::i r. II S 1. e:: q(s)ds} pes) u(1) ..

t

i=O e;i _r. ( ) 1 , 1.

This regular expansion does not satisfy the boundary condition u(l) .. 0 of problem (1.1), hence, we have to correct for this by a boundary layer

expansion. Substituting in the equation the local variable p:= (x - I)/E,

expanding the coefficients p and q in Taylor series,at x

= }

and inserting

the boundary layer expansion

I

EiS. we obtain the system of differential

1

equations (6

=

ds/dp)

(4.5a)

- s.

+ p(l)

s.

J J

=

-with the boundary conditions (4.5b)

s.(O)

a

-r.(I)

J J and _p+-<>o

lim

s.(p) ... J O.

We find

(4.6a) (4.6b)

and we see that s. is a polynomial of degree 2j in p multiplied by exp (p(J)p). J If S . is the solution of E,J (4.7)

o ,

u(l) ... -

!

i=O i E r.(l) 1

and if ;. (x) := s. «(1 - x) IE), the boundary layer expansion satisfies by (3.3)

1 1

the estimate

(4.8) $ CEJ + • 1 j+

r

1 Ir.(})1 $

. 0 1

1""

Since by definition R . + S . a Ur' formulae (4.4)& (4.8) yield the

E,J E,J 0:.

estimate

(4.9) IIU

-E

REMARK I: The highest order term Ej+] ; . 1 of the singular expansion

J+

L

Ei ;. in (4.8) is of the same order as the error estimate, hence, it can

1 . + 1

be dropped. However, its presence is necessary for proving the order O(EJ )

error estimate (4.10)

From the asymptotic approximations of U we can derive error bounds for the best approximation of U£ in the trial s:ace

E~

by construction of approximations to the regular and the singular part of the asymptotic expansion of U •

£

LEMMA 6: Let ~ E P satisfy the same boundary conditions as R 1· does,

£ I £,

~E(X) := i(r

O(I) + Erl(l»(x + I).

The linear manifolds

~E

+

E~

and

-~£

+

E~

contain the approximations

P~

and

h

a of Rand S respectively,

E £,1 £,1

They satisfy the estimates (4.1 ta) (4. lIb) _{II p} h _{- R III} £ £, E (4.llc) II a h - S )11

~

E -

~

II a h - S 1"

~

Ce: ill £11 .1 E £, I e: E, e: (4. lid)

for all E, h EO (0, 1] and k ~ 1.

PROOF: Since R I + S 1'" U , formula (4.1 td) is a consequence of .(4.11a & c).

E, e:, E

The estimates (4.lla & b) follow from (4.4) and well-known polynomial inter-polation. As approximation of the singular part we define for x E (xi_l,x

i) (4.12) ah(x) := -(reO) + £r(l) exp(p(l)(x. - I)/e:) (exp(p(x.)(x - x.)/e:) +

£ 1 · 1 1

x. - x

+ _1 _ _ _ {exp(p(1)(X

1'_1 - x1,)!e) - exp(p(xi )(x1''';'1 - Xi· )/£)}) •

xi - x_{i - 1}

Clearly this is an element of -cP + Ehk if k

~

1. In view of formula (4.10)

h e; I

term of its derivative. By the mean value theorem we find an intermediate

point ~. ( (x. l'x.) such that

1. 1- 1

Since

I

pel) exp(p(l)(x - x.)/e) - p(x.) exp(p(x.)(x - x.)/e)1

=

E 1 1. 1. 1

=

I

l(1 - x.)p'(;.)(l -(x - x.)/e) exp(p(;.)(x - x.)/e)1 s

E 1 1 . 1 . 1. 1.

X € (x. l'x.).

1. - 1.

this implies the estimate (4.llc), q.e.d.

If h is large in comparison with E, the boundary layer is contained

entirely in the subinterval (x 1,1) of the partition ~, hence, the

expone

n-tial trial functions in the other subintervals of the partition are super-fluous. We find:

LEMMA 7: inf IIU - vIII s C(e) + hk +

E-~e-h/e)(!lfli

+ II Dk+1 fll ),

h

V€E

k ,p

provided e s h s 1 and k ~ I.

h

PROOF: Instead of the singular part cr of (4.12) we choose

E (4.13a) Clearly it satisfies ifx l : s x s l , n-otherwise. (4.13b) _{II cr} h _{- SOli 1} .... E,p

-!

h ~

-i

-hie s e l l cr - s 1/ s C e ( e + e ) 1/ £II 1 ' e,p 0 E

provided E S h :s I. In conjunction with (4.11a) this implies the lemma, q.e.d.

An approximation of Green's function G (x,;) is constructed in an

e

analogous fashion by formal series expansions in powers of E.

As a function of ; it is the solution of (4.14a) L

*

u '" 0

(4.14b) u(x - 0) == u(x + 0), u'(x - 0) == U t (x + 0) + -1

E

(4.14c) u(-l)

=

u(l)

=

O.

*

Since the coefficient of u' in Luis negative, we expect that G (x,-)

E E

has boundary layers at ~ - x and at ~

=

-1. We begin with the construction of formal approximations to two linearly independent solutions of L:U - 0 which take the value 1 at ~ = x. It is natural to choose them such that one of them is approximated by a regular series and the other by a sing~lar

series only. An approximation of G is obtained by a suitable linear

E

combination of the regular and the singular expansion.

Inserting in L*u

=

0 the regular expansion

I

Eir.(x,~)

with the prescribed

E l ·

values rO(x,x)

=

1 and ri(x,x) == 0 (i > 0), we obtain by analogy to (4.2) the system of equations

=

0, rO(x,x)

=

1 and -(pr.)' + qr.

=

r'.' l' r.(x,x) == 0, 1 1 1- 1 resulting in (4.15)

J

i

q(tl - ,'(tl

_p(t dt} r.(x _' e) ==

~

" ( t) ( t) dt 1 ,..

J

r i _1 x, ro x,

p('tJ.

x x

The singular expansion is obtained by substituting in the equation the local variable. _{. x} := (~ - X)/E and inserting the formal expansion

1.

El s.(x,. ) with the prescribed values

1 1. X

s.(x,O) == 0 (i > 0),

1.

and the decay condition at infinity (outside the boundary layer)

lim s. (x,.)

=

O.

1

'-+<Xl

Collecting equal powers of E we find by analogy to (4.5a) a set of equations

which determine the functions s .• The zeroth and first order terms are

1.

(4.16)

By definition these regular and singular expansions satisfy the estimates, if ;.(x,~) := s.(x,(~ - X)/E), 1 1 m i m+l ilL

_L

E r. (x, ')II L2 ( ) =::; C e: e: i=O 1 -a,x m+l i m+l ilL

L

E _{'; i (x,}

~

_{)11 L2 (x, 1)} =::; C e: e:: i=O

...

From rO and

So

+ ESt we construc~_afirst order approximation of GE;

approximations of higher order are constructed analogously. A function which is equal to arO(x,.) for ~ E (-I,x) and to asO(x,.) + East(x,.)

for ~ E (x,I) is continuous on (-1,1) and its derivative has at ~

=

x the

jump

N(P(X) + p'(x) - 2q(x) )

~ E p(x) •

The choice a:= p/(p2 + Ep' - 2£q) yields an approximate solution of (4.t4a) which satisfies (4.14b) exactly. Adding to this function smooth terms in order to satisfy (4.14c) we find the first order approximation H of G ,

£ E

(4.18)

if x < f; < I,

+

if -I < x < f; •

Since G (x,-) and H (x,-) both satisfy (4.14b), their difference is in H2(-t,J).

£ £

Hence, we conclude from (4.17) and (3.6)

(4.19a) II G (x,·) - H (x,')11 s; ell L (G (x,·) - H.,. (x, ')11 S; Ce:,

e: £ e: E e: ~

and from Sobolev's inequality (3.3) we find (4.19b)

where C is independent of e: and x.

Analogously to lemma 1 we derive from this estimate error bounds for the b _{est approx~mat~on} . . ₀ f G . _{e: ln k:} Fh

LEMMA 8: The trial space Fkh contains an approximation Gh . of the Green's

£,l.

function, which satisfies the estimates (4.20a)

(4.20b)

for all E,h E (O,IJ, x. E d and k ~ J.

5. ERROR ESTIMATES FOR THE GALERKIN APPROXIMATIONS.

From the error estimates for the best approximations in the trial

spaces

E~, E~,p

and

F~

we derive error estimates for the Calerkin approximation

of the solution U of problem (1.1).

e

If the trial spaces are fitted exponentially to the singular solution

of Leu a 0, we obtain the result:

THEOREM I: The solution

u~

E

E~

of the problem (cf. 1.8)

(5. 1 )

satisfies the error estimate (5.2)

for all e; ,h E (O,IJ and k ~ _1.

Since U h satisfy (5. 1) , we have

PROOF: and U both

e: e: B (Uh e: e: - Ue:'v)

=

0, h V V E Ek •

Hence the estimates (3.7 & 9) imply

(5.3)

"" B (Uh - U , v - U )

~

C IIUh -

u

II II v - U II 1

£ e: £ £ £ E £ e

Minimizing this inequality over

E~

and using lemma 1 we find

(5.4)

In order to obtain a better bound we have to consider the regular and the

singular parts R£,1 and S£,) separately, as in lemma 6. Let ~e be as in

lemma 6 and let

both be solutions of (5.1). By linearity we have

u

h

=

Rh + sh.

E £ £

By analogy to (5.3-4) lemma 6 implies the estimates

(5.5a) IIRh - R II s C(e 3 / 2 + hk), £ £,1 £

(5.5.b) II Sh -

s

II S

c£

i ..

The second estimate can be improved if instead of (3.9) we use the inequality

(5.6) B (u,v)

=

e(u',v') + (u',pv) + (qu,v)

=

e:

=

2e;(u',v') + (u', - e;v' + pv) + (qu,v) ~

:s; cli ull II vii + II u' 1111 - e;v' + pvll • e; I::

From (4.8) and (4.12) we find

(5.7) II

(~d

- p)(oh -

s

1)11

~

ne:<2.... -

p)(Oh - s - I::S )11 +

x e; 1::, dx e;

°

1

3/2

+ II S - s - e;S 111.,. ~ Ce;, •

£,1

°

...

Formulae (5.6

&

7) imply

h .

II S - S 111:s; C e; I:: e;,

and in conjunction with (5.5a) this proves the theorem, q.e.d. COROLLARY: if e; :s; h2, partial

the solution Uh E Ekh of

e; ,p

h

B (U ,v)

=

(f,v) e; E

fitting yields the same error estimates;

satisfies V £ :s; h2 :s; 1 and V k

~

1 the estimate

IIUh -

u

II :s; C(I:: + hk)(lIfli + IIDk+1£II). I:: £ e:

Proof: Apply in the preceding

If the trial spaces are of the adjoint equation L*u

=

I:: THEOREM 2: The solution Uh €

e:

B (Uh,v) • (f,v)

e: I::

satisfies the error estimate (5.8)

proof lemma 7 instead of lemma 6,q.e.d.

fitted exponentially to the singular solution 0, we obtain convergence at the mesh-points only:

h

Fk of the problem

V v € Fh

k

V I::,h € (O,IJ,i=l, ••• ,n-l,k ~ 1.

Since U satisfies the same inequality,we find the a priori estimates

e:

(5.9) II

U~I

s II fll ,

e: e: II U II e: e: s II fll • Hence the error satisfies

(5.10a) II Uh - U 11

s

211 fll , e: e: e:

(5.10b) B (Uh - U ,v)

=

0,

e: E e:

1

As is well known, each u E HO(-l,l) satisfies the identity

(5. 11) u(x)

=

(L u, G (x,-» - B (u,G (x,-».

e: £ e: e:

Combining (5.]Ob) and (5.11) we find, cf. [IJ,

(5.12) _{Iuh(x) -}

u

_(x)1

₌

IB

_(Uh _{- U , G (x '»1}

=

£ E E £ £ £ '

=

IB

(Uh - U G

(x,-) - v)l s

£ e: E' e: s II U - Uh II II G (x, -) - vII 1 ' e: e: e: E

In conjunction with (5.]Oa) and lemma 8 this implies (5 _ ] 3)

In order to obtain the sharper estimate (5.8) we have to construct

approximations in

F~

to both the regular and the singular part of H£, cf. (4.18),

separately as in lemma 1; because of the discontinuity we have to do this on the subintervals (-l,x.) and (x.,]) separately_ The approximation of the

1. 1.

regular part is again of the order 0(£3/2 + hk) in the norms of H1(-1,x.)

. 1.

and H1(x.,I). The analogue of (5.6) is the estimate

1.

(5. 14) B (u, v) = (u', £v' + pv) + (u,qv)::;; II u '1111 EV' + pvll + II ulill qvll •

£

As in (5.7) we find that II e:v' + pv\1 is of the order 0(£3/2) i f v is

the difference of the singular part of Green's function and its approximation

by the exponentials of

F~,

because these exponentials satisfy the equation

EV' + p(x. l)v

=

0 on (x. l'x,), q.e.d.

1.- 1.- 1.

Combination of both ways of fitting yields globally, i.e. in the energy norm, the same result as in theorem 1, but at the mesh-points we obtain superconvergence, cf. [IJ.

h h h '

THEOREM 3: If Ue: E Ek + Fk is the solution of

(S. IS) B (U ,v) h

=

(f,v), e: e:

it satisfies the error estimates (S.16a)

(S.16b) (i

=

I, ••. , n-I ) ,

for all e:,h E (O,IJ and k

~

I. If e-h/e:

~

Ce:, we obtain the same result by

partial fitting (replacing

E~

by

E~,p)'

PROOF: The proof of (S.16a) is identical to the proof of (S.2). We obtain (S.16b) by using (S.16a) instead of (5.10) in the proof of theorem 2, q.e.d.

The results of these theorems are somewhat unnatural since either the test or the solution space or both spaces contain inadequate trial functions:

w~ is inadequate in the solution space, since it does not fit to the singular

solution of the problem and does not improve the best approximation of the

solution of (1.1) in the solution space; likewise

w:

does not improve

1

the best approximation of Green's function. In the previous theorems these inadequate trial functions have to be present, since these theorems are based on the a priori inequality (3.7), which requires the solution and

test spaces to be equal. If h + e:/h is sufficiently small, however,

these inadequate trial functions need not be present, when the error estimates

are based on the lemmas 4 and S.

THEOREM 4: If Uh E Eh is the solution of

e: k

(5. 17) B (U ,v) h

=

(f,v), e: e:

it satisfies the error estimates (5.18a)

(5.18b)

provided h + e:/h < y, where y as prescribed by lemma 4.

(i = I, .•. , n-I ) ,

PROOF: The estimate (5.18b) is proved from (5.18a) in the same way as (5.16b) from (S.16a) in theorem 3, and the proof of (5.18a) is almost the same as the proof of (5.2); differences arise in (S.3) and (5.6) only. In (5.3)

the error Uh - U seemingly is compared with the error of the best

tion of U 1n the test space. The following version, in which

U

h denotes

£ £

the best approximation of U in the solution space, yields a comparison to

£

the best approximation in the solution space:

(5.19) B (Uh - U v)

=

0 £ £ £' implies B (U h -

Uh,v)

=

B (U £ £ £ £ £ ""h - U

,v).

£

With the choice v:=

estimate for B ,

cf. (3.16), we find from lemma 4 the lower

£

(5.20) B (U - U ""h ,v) ~ ~ II U - U h ""h II II VII •

£ £ £ £ £ £ £

h

B is estimated from above for the regular and singular parts of U separately

£ £

as in the proof of theorem 1, but instead of (5.6) we use the estimate (5.21 )

q.e.d.

Finally we find that partial fitting of the solution space yields almost as good results as complete fitting, if £/h is small enough:

THEOREM 5: If Uh (Eh is the solution of

e: k,p

(5.22) B (U h ,v)

=

(f,v),

e: e:

v

V E ph k,p

then it satisfies the error estimate (5.23a)

and if v in (5.22) ranges over

F~_I,P'

then it satisfies (5.23a) and the

pointwise estimate

(5.23b) (i= 1 , ••• ,n-1) ,

-hie:

provided e < Ce: and h + e:/h $ y, where y is prescribed by lemma 5.

PROOF: The estimate (5.23b) follows from (5.23a) in the same way as before, the only difference being the fact that the degree of the polynomials in the test space is k - 1 and hence that Green's function can be approximated

k-l

to the order O(e: + h ) only. Formula (5.23a) follows from lemma 7 in the

TABLE 1

Survey of the orders of the error estimates obtained

[solution Test Dimension Order of the error restrictions

~pace space in II-Ii -norm at Mesh points in the proof e:

ph _{nk-l (*) 1 (*) 1}

-k k

f:

Eh _{nk+n-J e:+h}k _(*) k

k e:+h

-f~,p

Eh _k,p nk e:+hk (*) e:+h k e:::>h 2

~:

Fh _{nk+n-l J} k k e:+h

-~:+F~

Eh+Fh _{nk+2n-) e:+h}k _e:2_+h2k

-k -k +Fh ,p k Eh +Fh k,p k nk+n e:+h k e:2+h2k e:-<h2

~:

_Pk+l h nk+n-l e:+hk (*) e:+hk h+e:/h::>y

h _ph

e:+hk e:+hk e:::>h2

f~,p

_k,p nk (*)

r:

Fh _{nk+n-l e:+h}k _e:2_h2k _h+e:/h::>y k h _Fh e:+hk 2 h2k <h2 F;,p k,p nk e: + e:-~h Fh _nk+n-l

_(**)

)

_(**)

2 h2k+J h+e:/h::>y I"k+l k e: +

Remarks: The results, marked by (*) or (**) are not proved in the theorems

stated above, they are mentioned for reason of completeness. The results

marked by (*) follow easily from the fact that the solution of (1.1) and the

aeen1s function are of order unity with respect to the 11·11 -norm, cf.

(5.9).

. e:

The remarkable result

(**)

reflects the fact that the O(l)-error in the

11·11 e: _{-norm is cOIllllitted almost completely in the subinterval (xn_} 1'1), where the boundary layer is located and that the Green's function is small in that

LITERATURE •

[IJ J. Douglas & T. Dupont,

GaZerkin approximations for the two-point

boundary probZem using oontinuous, pieoewise poLynomial spaoes,

Numer. Math. ~ (1974), p. 99-109.

[2J W. Eckhaus,

Matohed asymptotio expansions and singular perturbations,

North-Holland Mathematics Studies 6, North-Holland publ. co., Amsterdam, 1973.

[3J P.P.N. de Groen,

Singularly perturbed differential operators of seoond

order,

Math. Centre Tract 68, Mathematisch Centrum Amsterdam, 1976,

ISBN 90 6196 120 3.

[4J P.P.N. de Groen,

The nature of resonance in a singular perturbution

problem of turning point type,

to appear in SIAM J. Math. Anal.,

(preprint: Memorandum 1977-003, Dept. of Math., University of Technology, Eindhoven, The Netherlands).

[5J P.W. Hemker,

A numerioal study of stiff two-point boundary probZems,

Math. Centre Tract 80, Mathematisch Centrum Amsterdam, 1977,

ISBN 90 6196 146 7.

[6J R.E. O'Malley,

Introduotion to singuLar perturbations,

Academic press, inc., New York - London, 1974.

[7J P.P.N. de Groen & P.W. Hemker,

Error bounds for exponentially

fitted Galerkin methods applied to stiff two-point boundary

vaZue problems, to appear in the prooeedings of the oonferenoe

on numerioaZ anaZysis of singUlar perturbation problems,

Nijmegen (the Netherlands) 30 May - 2 June 1978, eds. P.W.