I

CAN A FINITE ELEMENT METHODPERFORM ARBITRARILY BADLY?

Ivo BABUSKA AND JOHN E. OSBORN

ABSTRACT. In this paper we construct an elliptic boundary value problems whosestandard finite element approximations converge arbitrarily slowly in the energynorm, and show that adaptive procedures cannot improve this slow convergence. Wealso show that the L2-norm and the nodal point errors converge arbitrarily slowly.With the L2-norm two cases need to be distinguished, and the usual duality principledoes not characterize the error completely. The constructed elliptic problem is onedimensional.

1. INTRODUCTION

The classical finite element method approximates the exact solution U of an el-liptic boundary value problem by piecewise polynomials (or pull-back polynomialswhen curvilinear elements are used). Denoting by UN the finite element approx-imation with N degrees of freedom (i.e., the approximate solution for which Nunknowns have to be determined), for reasonable meshes we have Ilu - uNIlE ~ 0(II· liE denotes the energy norm) as N ~ 00, provided U E HI, If u has additional

:=Eosmoothness, then typically Ilu - uNllE ~ CN n , where µ depends on the smooth-ness of u and the degree of the elements, and n is the dimension of the problem. Ifthe solution u has additional properties, e.g., u is the solution of Laplace's equationin a polygonal or polyhedral domain and is singular at the corners or edges, and themesh is properly selected a priori or by an adaptive procedure, then we typically

.=..ehave Ilu - uNllE ~ CN n , where p is the degree of the elements.Hence the question arises whether there are problems for which the classical finite

element method converges arbitrarily slowly. We show there are such problems,and furthermore show that the convergence cannot be improved with adaptivity.Specifically, we show that given a sequence of nonincreasing positive numbers XN,with Xl = 1, that converges to 0, there is a problem with solution u E HI such that

• there are constants Cl and C2, independent of {XN}, such thatCIXN ~ Ilu - uNIlE ~ C2XN, for all N

(briefly, the energy norm error is of order XN);• there is a sequence 1 ~ Nl < !V2 < ." such that Ilu - UN; IIL2 is the order

of XJv;' provided XN converges to 0 "very slowly" (in a sense to be mademore precise later); the usual duality principle does not characterize the errorcompletely;

and• the nodal point errors are the order of xJv.1991 Mathematics Subject Classification. Primary: 65N15, 65N30.Key words and phrases. Finite element methods, convergence, adaptivity, rough coefficients.The first author was supported in part by NSF Grant #DMS-95-01841

These results are proved for a family of uniform meshes. We also show the errorsbehave the same way for adaptively constructed meshes.

The example we construct is a one-dimensional boundary value problem, with ahomogeneous differential equation, a homogeneous Dirichlet condition at one end,and a non-homogeneous Neumann condition at the other end. The differentialequation has a rough coefficient a(x), satisfying 0 < a ~ a(x) ~ (3, which isconstructed in terms of the sequence {XN}. We expect that the results we obtainalso hold for general boundary value problems: boundary value problems in one ormore dimensions, with general Dirichlet, Neumann, or mixed Dirichlet/Neumannboundary conditions,

Problems with rough coefficients are typical in problems with heterogeneousmaterials. The example we construct shows that finite element methods based onpiecewise polynomial elements lead to unacceptable results when applied to theseproblems. In addition, it shows that the comparison of results computed withcoarser and finer meshes cannot be used as a basis for assessing the accuracy of theresults. This example also shows the importance of developing special methods forthese problems; the special element method introduced and analyzed in [2, 3] is anexample of such a method.

Section 2 describes the specific boundary value problem we consider. In Section3 we construct the specific coefficient a(x), and state and prove the estimates forthe energy norm of the error. The estimates for the nodal point errors are provedin Section 4. In Section 5, the L2-error is analyzed. Remark 7, in Section 5,discusses the relation of the L2-error to L2-error estimates proved with the usualduality argument. Remark 8, in Section 5, discusses the error in the Loo-norm andsuperconvergence at the nodes. Section 6 discusses adaptively constructed meshesfor our example. Section 7 summarizes our conclusions.

2, A MODEL BOUNDARY VALUE PROBLEMAND ITS FINITE ELEMENT ApPROXIMATION,

We consider the specific model boundary value problem

(2.1){

d du- dx (a (x) dX: = 0, 0 < x < 1

u(O) = 0, au (1) = 1,

where a(x) is a measurable function on I = (0,1) satisfying

(2.2) o < a ~ a(x) ~ (3,

The solution u(x) of (2.1) can be interpreted as the longitudinal displacement ofa heterogeneous bar with (local) modulus of elasticity a(x) that is subject to nolongitudinal load, with left end fixed, and with the stress at the right end equal to1.

We will understand (2.1) in the weak or variational sense:

(2.3){

u E HJ,(I)

B(u,v) =11

a(x)u'v'dx = v(1) Vv E HJ,(I),2

where

andHJ,(1) = {u E H1(I) : u(O) = O}.

(The superscript l indicates that the functions in HJ, (1) are required to be 0 at theleft endpoint of the interval I.) The solution of (2.3), or (2.1), exists and is unique,and is given by

(2.4)(X dt

u(x) = Jo a(t)'

Sometimes we will indicate the dependence of u on the coefficient a by writing Ua.

On HJ, (1) we will also use energy norm

11 1/2IluliE = [B(u, u)F/2 = [ 0 a(u')2dx]

and the norm

11 . 1/2IUIHl(I) = [ 0 (u')2dx] .

IluliE and lulHl(I) are equivalent on HJ,:

We will also use the L2(I)- and Loo(I)-norms.We are interested in the approximation of the solution of (2.3) by a usual (poly-

nomial based) finite element method. Toward this end we let ~ = {O = x~ < x~ <... < x~ = 1}, where N = Nt:. is a positive integer, be an arbitrary mesh on I,and let Ij = If = (Xf-l' xf), j = 1, ... ,N, and h = hA = maxl~j~N(xf - Xf-l)'Further, let

(2.5) VA = {u E HJ,(I): Ulra E pl(It),j = 1, ... ,Nt:.},]

where pI (If) is the set of polynomials of degree less than or equal to 1 (consideredon If), be the associated finite element space. Then, as usual, the correspondingfinite element approximation ut:. to the solution u of (2.3) is characterized by:

(2.6)

Clearly, uA exists and is unique.It is known that

(2.7a) lim lIu - ut:.IIE = 0,ha..-rO

3

and hence that

(2.7b)

provided a(x) satisfies (2.2), and furthermore that Ilu - ut:.IIE = O(ht:.) if a(x) issmooth. We are interested, however, in rough a(x), and in assessing the accuracyof uA for such a(x). We will see that essentially nothing beyond (2.7a,b) is truewithout additional assumptions on a(x).

Throughout most of the paper we will consider uniform meshes. Specifically,consider the family of uniform meshes given by

o 1 2k~ = ~k = {Xk,O = 2k = 0 < Xk,l = 2k < ... < xk,2k = 2k = 1}, k = 0,1, ... ,

and letVk = Vt:.k.

Then N = NAk = 2k;Ij = Ik,j = Itk = (Xk,j-bXk,j),j = 1,... ,2k; and h =hk = hAk = N-1 = 2-k. The corresponding finite element approximation Uk to uis characterized by

(2.8)

cf. (2.6). Note that degrees of freedom = dim Vk = N = 2k.

3. ANALYSIS OF THE ERROR IN THE ENERGY NORM.

Let u be the exact solution of the boundary value problem (2.3), or (2.1), and letUk be the finite element approximation determined by the uniform mesh ~k with2k elements and hk = 2-k described in Section 2. In this section we assess the errorin the energy norm,

Theorem 1. Let {Xk}k'=01 with Xo = 1, be a sequence of nonincreasing positivenumbers converging to O. Then there is a coefficient a(x) satisfying (2.2) (witha = 0.9 and (3 = 1.1) such that such that

(3.1)

Remark 1. We have chosen to present the constants in our estimates as explicit(decimal) numbers. We have adopted this somewhat unusual practice in part toshow that the constants are absolute, in particular that they do on not depend onthe sequence {Xk}, and in part to make the proofs of the estimates easier to follow.The coefficient a(x) naturally depends on {Xk}.

Remark 2. The sequence {Xk} is indexed by k and not by N = degrees of freedom.The result can, however, be recast in terms of N using the relation N = 2k; cf.Section 1. The same is true for Theorems 2 and 3 below.

4

..

Proof.Let Uk be the approximate solution, as characterized by (2.8). Writing

2k

Uk = 2:: Uk,i<!)i,i=l

I adxLetting ak . = Ik'j and Zk . = (Uk,j-Uk,j-dak.j equations (3.2) can be written,) hL ,) h, ,

Zk,j - Zk,j+l = 0, j = 1, ... , 2k - 1, Zk,2k = 1.

Hence Zk,j = 1 for j = 1, ... , 2k, and equations (3.2) reduce to

(Uk,j - Uk,j-l) ak,j = l,j = 1" .. , 2k,

and so, since Uk,O = 0,

(3,3)

Let

From (2.4) we see that

(3.4)

We will define a(x) in terms of the following L2(I)-orthonormal sequence, whichis closely related to the Haar basis (cf. [10]):

5

go(x) = 1,

91(X) = { 1,-1,

0<x<1

O<X<~~< x < 1,

and, in general,1, O<x<ir-1, 1 22T<X<2T1, 2 32T<X<2T

gz(X) = ~ -1,3 42T<X<2T

1, 2'-2 < < 2'-12lX212'-1-1, 21 < x < 1.

Let, = 1/20, Then let 0 = lo < h < ." be inductively selected so that lHI is theleast index ~ li + 1 such that

(3.5a)

It is immediate that

(3.5b)

We then define a(x) by

(3.6)

where

00

a(x) = 2:: d1gl(X),1=0

(3.7) dlo = do = 1; dl; = ,Xli-1 for i = 1,2, ... ; and dl = 0 for l =1= lo, h,···,

We see immediately that dl; ~ ,iXO = ,i, and hence that

(3.8)

Hence the series defining a(x) converges uniformly and

(3.9) a = 0.9 ~ a(x) ~ 1.1 = (3.

The series defining a(x) is of lacunary, or gap, type.If 1 ~ k, we see that 91(X) is constant on each h,j,j = 1, ... , 2k, Hence, if we

write

•

(3.10)k 00

a(x) = Ld1gl(X) + 2:: dlgl(X)1=0 l=k+l

= 4>k(X) + TJk(X),6

we see that ¢k(X) = ¢k,j is constant on each h,j,j = 1, .. " 2k. Furthermore, ifl > k, then

(3.lla)

if l ~ k, then

(3.llb)

/, gl(x)dx = 0, for j = 1, .. " 2k;h,j

if l,m > k,l i= m, then

(3.llc)

and if I ;::::0, then

(3.lld) { 9f(x)dx = hk, for j = 1, ... , 2k.lh.j

(3.12a)

From (3,10) and (3.lla,c,d) we see that

{ 1Jk(x)dx = 0lh,jand

(3.12b)

We also have (cf. (3.8) and (3.9))

(3,13)

and

(3.14)

a = 0.9 ~ ¢k(X) ~ 1.1 = (3

(3.15);

From (3,10) and (3,12a), recalling that ¢k(X) = ¢k,j is constant on each Ik,j, wesee that ak,j = ¢k,j' Thus, from (2.4), (3.3), and (3.10), for x E h,j we have

du dUk 1 1dx - dx - a(x) - ak,j

1 1¢k,j + 1Jk(X) - ¢k .,)

_ -1Jk(X)¢~ .(1 + 77k(X»),) cPk ,j

= ¢~ . ( -1Jk(X) + 7j;(k,j,x) 1J~(X)),) A. '7 ~kJ

where, as a consequence of (3,13) and (3.14),

(3.16) I~(k, j, x) I ~ 1.2.

From (3.9) and (3,15) we have

[ /, (du dUk) 2 ] 1/2

a(x) d-T dx ~ Jalu - UkIHl(h,j)h,j x x

via [( /, ) 1/2 1 ( /, ) 1/2](3.17) ~ ¢>~,j Ik,j .,.,~(x)dx - ¢>k,j Ik,j ~2(k,j,x)'TJ:(x)dx .

Using (3.13), (3.14), and (3.16) we see that

1 ( /, ) 1/2 ( /, ) 1/2¢>k,j Ik,j ~2(k,j,x).,.,:(x)dx ~ 0.14 Ik,j .,.,~(x)dx ,

Thus from (3.12b), (3.13), and (3,16) we have

[1 (du dUk ) 2 ] 1/2 (1 )1/2 1/2 (00 ) 1/2I . a(x) dx - dx dx ~ 0,6 I . .,.,~(x)dx = 0.6hk 2:: dr ,

k~ k~ I=k+l

and hence

,

(3.18)00 1/2

lIu - ukllE ~ 0.6( 2:: dr) , for all k.l=k+l

For k = 0, 1, . ,. let i be such that li ~ k ~ li+l - 1. Then, using (3.7) we have

(3.19)00 00 00

"'" d2 = "'" d2 = ",,2 "'" X2 > ",,2X2. > ",,2X2.~ I ~ 1m I ~ 1m - I I, - I kl=k+l m=i+l m=i

Combining this estimate with (3.18) we get

(3,20)

which is the first estimate in (3.1).

Remark 3. It follows from the definitions of a(x), the mesh family {~d, and thefinite. element approximation Uk that

By a similar argument we get an upper estimate. In fact, we get

8

..

Now, with li ~ k ~ l(i+l) - 1, using (3.5a), (3.5b), and (3.7) we have

(3.22)

, Combining this estimate with (3.21) we get

(3.23)

which is the second estimate.in (3.1).

Remark 4. It is possible to base Theorem 1 on a standard result in approximationtheory. It is known [8, 16] that given any sequence {xd, there is a U E HJ, (1)such that inf¢>EvkIu - ¢IHl = Xk for all k, i. e., there is a function u with specifiedapproximation properties. Now, we are interested in a solution u that correspondsto a coefficient a(x) : U = Ua. It immediate from (2.1) that a(x) must be given by

(3,24) a(x) = ~- , ,

;

It is clear that a(x), defined by (3.24), satisfies (2,2) if and only if du(x)/dx· isbounded away from 0 and 00. Now, it is not clear that the function u constructedin [16] is in Loo, but with an alternate construction by p, Oswald [13], this isclear. Then by considering u(x) = u(x) + cx, for an appropriate value for c, weget a solution u(x) that has specified approximations properties, and such thatthe corresponding a(x) satisfies (2.1). In this way we obtain an alternate proof ofTheorem 3,1.

We have chosen to prove Theorem 3.1 as we did, however, because it leads natu-rally to the proofs of Theorems 2 and 3 below. We thank Peter Oswald for pointingout the approximation theory result [8, 16] and the construction [13] mentionedabove.

Remark 5. With Xk = 2-2k, Theorem 1 shows that Ilu - ukllE is of order Xk =2-2k = h~. This might seem to contradict the well known result that the highestpossible rate of convergence with piecewise linear elements is O(h), which is provedusing the theory of N-widths (see, e.g., [14]) and a saturation theorem (see, e.g.,[9]). But the theory of N-widths is concerned with the worst possible case, andsaturation theorems assume sufficient smoothness, so there is no contradiction,

4. ANALYSIS OF THE ERROR AT THE NODAL POINTS.

We again let u be the solution of (2.3) and let Uk be the finite element ap-proximation determined by ~k' In this section we assess the error at the nodalpoints,

Theorem 2. Let {Xdk=01 with Xo = 1, be a sequence of nonincreasing positivenumbers converging to O. Assume that the coefficient a(x) is defined by (3.5a),(3.6), and (3.7). Then

(4.1) (1.5· 1O-3)X~xk,j ~ I(u - Uk)(Xk,j)1 ~ 1.7X~Xk,j,j= 1,2,.,., 2k, for all k,9

Proof. We begin with a refinement of equation (3.15): For x E h,j,

(4.2)

(4,3)

where 1/J satisfies (3.16). Thus, using (3,12a), we have

{ (dU _ dUk)dx= -i-[ ( rJ~(x)dx- Ilk,j 1/J(k,j'~)rJ~(X)dX].1h,j dx dx ¢Jk,j 11k,j ¢Jk,)

:Now, using (3.13), (3.14), and (3,16) we have

IIh .1/J(k, j, x)rJ~(x)dx I 1,] ~ 0.14 rJ~(x)dx.

¢Jk,j h,j

Thus, from (3.12b) and (3.14) we have

(4.4)

Hence

{ (dU_dUk)dX~O,6 ( rJ~(x)dx11 dx dx 11k,jk,j

00

= O.6hk 2:: d;, for all k.l=k+l

00

= O,6( L dr)Xk,l,l=k+l

00

~ 0.6( L dl) (Xk,l + hk)l=k+l

00

= 0.6( L dl)Xk,2'l=k+l

..

and, in general,

00

(4.5) (U-Uk)(Xk,j)~0.6( 2:: dr)xk,j,j=1, ... ,2k, forallk.l=k+l

10

i

,

Combining (3.19) and (4.5) we get

(4.6) (u - Uk)(Xk,j) ~ 0.6"Y2X~Xk,j = (1.5 . 1O-3)X~xk,j,

which is the first estimate in (4.1).A similar argument yields

00

(u - Uk)(Xk,j) ~ 1.6( 2:: df)Xk,j'l=k+l

Combining this estimate with (3.22) yields

(4.7) (u - Uk)(Xk,j) ~ 1.7X~Xk,j,

which is the second estimate in (4.1).

Remark 6. From the usual finite element error analysis, we know that the nodalpoint errors are 0 if a(x) = constant, and are O(h2) if a(x) is smooth. We note thatneither of these results apply to our example, Note that we obtained arbitrarily lowrates of convergence by appropriately selecting the sequence Xk; the correspondingcoefficient a(x) is nearly constant (cf. (3.9)), but is not smooth.

5. ANALYSIS OF THE ERROR IN THE L2-NoRM

The usual duality argument [1, 11, 12] shows that

(5.1a)

where

(5.1b)

Wcf>being the solution of

{

d dwcf>- dx (a (x) dx ) = </>, 0 < x < 1

wcf>(O) = 0, aw~(l) = O.

If a(x) is smooth, specifically if a(x) E Cl[O, 1], then 1J(h) ~ Ch, and thus

(5.2) Ilu - ukIIL2(I) ~ Chllu - ukllEand

(5.3) Ilu - Uk IIL2 (I) ~ Cllu - ukll~·Estimate (5,3) follows from (5.2) and the estimate Ilu - Uk liE ~ Ch, which is validunder a mild hypothesis on u (in addition to smoothness) [6]. But if a(x) is rough,if we are assuming it is merely measurable, then, although limh..-rO 1J(h) = 0, noestimate of the form 1J(h) ::::;ChP, with p> 0, may hold. In this situation, we mayknow [5, 15] only that

(5.4)

In this section we derive estimates on the L2(I)-error. Following their derivation,we discuss their relation with L2-estimates derived via duality. We will also stateestimates on the Loo-error, and discuss their relation with superconvergence at thenodes.

11

Theorem 3. Let {Xdk=O' with Xo = 1, be a sequence of nonincreasing positivenumbers converging to O. Assume that the coefficient a(x) is defined by (3.5a) and(3.6), and (3,7). Then

(5.5) max{ (1.4· 1O-3)Xli-lhli-l, (0.53 . 1O-3)Xf;_1 - 0.36Xl;-lhl;-d~ Ilu - ul;-II1L2(I)

~ 1.7Xfi-l + 0.5Xli-lhli-l, for all i ~2.

Proof. Using (4.2) with k = Ii - 1, for x E !z;-l,j we have

(5.6)

(u - Uli-d(x) = (u - Ul;-l)(Xli-l.j-l) +1x

[u'(t) - UL_l(t)]dtX';-1.j-1

1 1x

= (u - Uli-d(Xl;-l.j-d - 12. . TJli-1(t)dtl;-I,) X';-l,]-l

1 1.x+ '3. . TJl_l(t)dt

li-l,) X';-1,J-1

1 1.x

- 4. 1/J(li-l,j,t)TJ~_l(t)dt.l;-l,) X';-1,j-1

We will estimate each of the terms on the right side of (5.7) in turn.12

..

,

Let

The graph of GI; (x) on the interval II;-I,j is shown in Figure 1.

'\-1,1-1E

Figure 1A direct calculation shows that

'\_1,):::0.

x

(5.8)

I

Now

is minimized by

c = c = JI,;_l,j GI; (x)dxhl;-l

Thus, another direct calculation yields

[(I <P~ . 12] 1/211 . GZ;(x) - d ,) (u - UI;-l)(XI;-l,j-l) dx ~

II; -1,) I,

13

Hence, using (3.7) and (3,13), we have

(5.9)

The graphs of Gl; (x) and Gli+1 (x) on the interval h-l,j are shown in Figure 2.

GJ (x) and GJ (x)I ~,

~I'I-',jhl(~,)_l

E

Figure 2Another direct calculation shows that

and, in general,

1IIGli+s IIL2(1,;-1,j) ~ 2sliGIi IIL2(1'i-1,i)'

14

x

...

,

So, using (3.5a), (3.7), and (5.8), we have

00 {X 00L dim lilT. glm (t)dtIlL (1.) = L dim IJG1m IIL2(1,;-1,j)m=i+l XI;-I,j-l 2 1,-1,] m=i+l

00

~ JIGI; IIL2(1,;-I,j) L 2~:im=i+l

00

= IIGI; IIL2(1,;-I,j) 2:: '~~:1m=i+l

200m

~ IIGI; IIL2(1,;-I,j)' ~;-1 2:: (i)m=O

Hence

(5.10a)

,

In a similar way we get

this estimate will be used later.Using (3.5a), (3.7), and (3.12b) we have

(5.11) 1 11x

17~_1(t)dtI2 dx ~ 1 11 17~-l(t)dtr dx1';-l,j X';-l,j-1 1';-l,j 1';-I,j~(1 17~_1(t)dt)2hl;_1

I,; -l,j00 2

= h?;_l (2:: dL)m=i

00 2

= h?;_l ( L ,2Xfm_1)

m=i

Hence

(5,12) c ~ (3.5 . 1O-3)h:!~lXf;_1'15

Using (3.14), (3.16), and (5.11) we get

! .11x

1/J(li - 1,j, t)7J~-l (t)dtI2 dx1'i-1,j-l X'i-1,j-l .

~ (0.12)211 lx

7J~-1 (t)dtI2

dxI,; -l,j -; x,; -1,j-1

(0.12)2h3 4

~ 399 1;-lXl;-l'

Hence

(5.13) D < (4.6· 1O-4)h3/2 X2 ,- 1;-1 li-1

Finally, combining (5.7), (5.9), (5.10a), (5.12), and (5,13) we get·

(5.14)

where in the last inequality we have used the fact that li-l ~ li - 1, which impliesXl;-1 ~ Xl;-I' It follows immediately from (5.14) that

(5,15)

We now prove another lower bound. From (5,6) we have

From (4,6) with k = li - 1 we have

"

(5.18) , . -3) 2 . .hl/2.A ~ (1.5 10 Xl;-lXl,-l,) li-l16

From (3.5b), (5.10b), (5.12), and (5.13) we have(5.19)B' + C + D ~ (2.4· 1O-2)h~/~lXl;_1 < (2.4· 1O-2)h~/~1,-lXli_l ~ 0.5h~!~IXl;-I'

Combining (5.17), (5.18), and (5.19) we have

Now consider intervals h-l,j such that Xl;-I,j-l ~ 1/2, i.e., consider j's satis-fying 21;-2 + 1 ~ j ~ 21;-1. There are 21i-2 such j's ifi ~ 2, and for each of themwe have

lIu - Uli-lIIL2{I,;_1,j) ~ [(1.5 '10-3)~Xfi_l - 0.5hli-lXl;-1]ht/~1'

Suppose (0.75· 1O-3)Xl;_1 - O,5h1;-1 > O. Then·

lIu - Uli-lIIL(I) ~ Ilu - Ul;-11IL(1/2,0)

2:: Ilu - Ul;-1IlL{I'i_1,j)2';-2+1~j~2';-1

~ [(0.75. 1O-3)XT;_1 - O,5h1;-lXl;-1 f h1;_121i-2

= ~[(0.75 ·1O-3)Xf;_1 - 0.5h1;-lXl;-lf, for i ~2.

Hence

(5.20)

We easily see that (5.20) is valid without the assumption that (0.75· 1O-3)Xl;_1 -O,5h1;-1 > O. Estimates (5.15) and (5,20) prove the first estimate in (5.5).

Now we prove the second estimate in (5.5). From (5.6) we have

(5.21)

Estimate (4,7) with k = li - 1 yields

(5.22) , 2 hl/2A ~ 1.7Xli-l Ii-I'

,

Combining (5.19), (5.21), and (5.22) yields

ThusIlu - Ul;-11IL2(I) :::;1.7XT;-1 + 0.5Xl;-lhli-l,

which is the second estimate in (5.5).

Remark 7, It is informative to consider the following two cases:Case 1. The sequence Xk converges rapidly to 0;

17

Case 2. The sequence Xk converges slowly to 0Suppose we are in Case 1. Then the l~s are not so large (we could, e.g., have

li = i), and XI;-l ~ hl;-l. Thus from (5.5) we see that IIu - UI;-11IL2(I) is of orderhl;-lXZ;-l. Combining this result with (3.1) shows that

(5.23)

Suppose next we are in Case 2, which we are mainly interested in. Then the l~sare very large, so XI;-1 » hl;-l. Thus (5.5) shows that Ilu - uI;-IiIL2(I) is of orderX~-1 Combining this result with (3.1) shows that

(5.24)

We can relate these results to estimates (5.2), (5.3), and (5,6). We see that inCase 1, an estimate of type (5.2) holds, but not (5.3). In Case 2, an estimate oftype (5.3) holds, but not (5.2), The duality argument used to prove (5.2) and (5.3)is, of course, not valid for our example, since it is not smooth. We have given directalternate proofs of (5.23) and (5.24).

Although (5.3) is proved only for smooth problems, if it is formally consideredfor a rough problem, it is similar to (5.4) in that the upper bound is a quantitythat goes to zero-possibly very slowly-times Ilu - ukllE. In this sense (5.24) iscompatible with (5.4).

Finally we note that although our example is rough in both case, Case 1 doeshave the following "smoothness": The series defining a (x) is "less lacunary", soa(x), and hence u(x), is "smooth".

Remark 8. With an analysis similar that used in the proof of Theorem 3, one canobtain bounds for the Loo-error, These bounds have the same form as those in(5.5). Specifically, one can show that

(5.25)max{CIXI;-lhl;-1,C2Xf;_1 - C3XI;-lhl;-d ~ Ilu - UI;-IIILoo(I)

~ C4Xf;-1 + CsXz;-lhz;-l'

In Case 1 (see Remark 7), (4,1) and (5.25) show that the Loo-error is of orderhz; -1 Xl; -1, and that the nodal point errors are of the higher order Xf;-1' establishingsuperconvergence at the nodes. This is the error-behavior we expect with a smoothproblem. In Case 2, the nodal point errors and the Loo-error are of order Xf;-l'showing that there is no superconvergence at the nodes, This is the error-behaviorwe expect with a rough problem.

Remark 9. The same coefficient a(x) is used in all three theorems. To be precise,with 'Y = 1/20 and with a(x) defined in (3.5a), (3.6), and (3.7), estimates (3.1),(4.1), and (5.5) for the energy-norm error, the nodal point errors, and the L2-error,respectively, simultaneously hold,

6. ADAPTIVITY

So far we have worked with uniform meshes. Consider now a family of mesheswith nodes of the form j2-k, with j taking on a sequence of N + 1values between

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..

o and 2k, inclusive, Such mesh families are often constructed by adaptive proce-dures, and in most practical situations these adaptive procedures produce meshfamilies and associated approximate solutions whose rate of convergence, measuredin the energy norm, is O(ljN). Suppose we use the following, typical, adaptiveprocedure. Starting from a uniform mesh, we consider the energy norm error oneach subinterval of the mesh. We then refine, by dividing in two equal parts, eachsubinterval whose error is greater than or equal to 8x (maximum subinterval error),where 0 < 8 < 1 is a specified parameter. Following this refinement, we repeat theprocess. The resulting mesh will in general depend on the solution u(x), and thuson the coefficient a(x). We denote the meshes by ~N' This approach is based onthe equilibrium principle, which tries to make the errors in the elements approxi-mately equal. This principle is used in all adaptive approaches. For an analysis of

. this and similar approaches, we refer to [4, 7].We examine this adaptive process for our boundary value problem, (2.1), with

the coefficient a(x) defined by (3,5a), (3,6), and (3.7). We claim that if 8 < ~, andwe start with the uniform mesh ~o = {O,I}, then the adaptive procedure producesa uniform mesh family. To prove this, suppose that at some stage we have theuniform mesh ~k = {O,2-k, •.• , I}. It follows from (2.4), (3.3), (3.9), (3.10), and(3.13) that

{ al(u _ uk)'12dx = ( . lak,j - a;x) 12

dx1 1k,j 11k ,) a(x )ak,j

~ (0.9)-31 lak,j - a(x)12dxh,j

= (0,9)-3 ( lc/>k(X) - a(x)12dx11k,j

= (o.9)-31k,j l17k(X)12dx

and

1 al(u - uk)'12dx ~ (1.1)-31 lak,i - a(x)12dxIk,i h,;

= (1.1)-3 ( l17k(X)12dx.1h,;

Now, it follows from (3.12b) that

lk,j l17k(x)12dx = lk'i l17k(x)1

2dx.

Hence,

1 al(u - uk)'12dx ~ (-~:~) 31 al(u - uk)'12dx ~ 2 { al(u - uk)'12dxIk . 0.9 1k . 11k',3 ,t .'

for any two subintervals Ik,j and Ik,i of ~k' Thus, if 8 < ~, every subintervalis subdivided, and the refined mesh is ~k+l' So, starting with ~o, the adaptiveprocedure produces ~l, ~2,""

Since the adaptive procedure will produce only uniform meshes, we see that theerror in the energy norm, the nodal point errors, and the error in L2 are as indicatedin Theorems 1, 2 and 3.

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7. CONCLUSIONS

a) We have shown that there are problems whose finite element approximationsconverges arbitrarily slowly, and that adaptivity cannot improve this situation.Specifically, if Xk converges to 0, with xo = 1, then there is a problem such that• the energy norm error is of order Xk;• the nodal point errors are of order X~; and• lIu - UI;-IIIL2(I) is of order h1i-lXI;-1 if Xk converges rapidly to 0, and is of

order Xfi-l if Xk converges slowly.Adaptivity does not improve the convergence.

b) The cases Xk ~ 0 rapidly (Case 1) and Xk ~. 0 slowly (Case 2) need to bedistinguished; they lead to different behaviors, The relation between the L2 (1)-error and the energy norm error is compatible with the smooth problem estimate,

in Case 1. And it is compatible with smooth problem estimate,

in Case 2.c) We have shown that we have superconvergence at the nodes in Case 1, but not

in Case 2.

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TEXAS INSTITUTE FOR COMPUTATIONAL AND ApPLIED MATHEMATICS, UNIVERSITY OF TEXAS

AT AUSTIN, AUSTIN, TX 78712

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MARYLAND, COLLEGE PARK, MD 20742E-mail address: jeo<Omath.umd,edu

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