TICAM REPORT 96-04February 1996

The Babuska-Brezzi Condition and thePatch Test: An Example

I. Babuska and R. Narasimhan

THE BABUSKA - BREZZI CONDITION AND THE PATCH TEST:

AN EXAMPLE

LBABUSKA AND R. NARASIMHAN

ABSTRACT. This paper discusses an example of a mixed finite element method in one-dimension which satisfies the engineering patch t.est, but does not sat.isfy a uniform inf-supcondition. The inf-sup constant for this method is shown t.o be of order h2. The reasons forgood results from benchmark problems for this method are addressed by characterizing thenature of input data for which the method performs well. Numerical results illustrating theeffect of the non-uniform inf-sup condition are presented.

1. INTRODUCTION

The Babuska-Brezzi uniform inf-sup condition is accepted as a necessary condition for

an effective finite element method. Since the analysis of the inf-sup condition is not always

straightforward, various simplified conditions are often substituted. For example, some

finite elements used in engineering practice are ?-dvocated because they pass the patch test

criterion [1], [2], [3]. These methods are tested numerically on a certain set of benchmark

problems and yield good results. It is possible that these methods may not necessarily sat-

isfy the uniform inf-sup condition, and thus a contradiction arises between the conclusion

from the mathematical theory and the results of some engineering practice.

In order to clarify this contradiction, it is important to understand that the uniform

BabuSka-Brezzi condition is a necessary condition for optimal performance of a method

when it is applied to a well-defined set of the input data A. Such a method is said to

be robust with respect to the set A. When the uniform inf-sup condition is not satisfied,

there exist input data in A for which the method gives suboptimal or no convergence. In

this case, the method is not robust with respect to the input set A. Nevertheless, it is

possible that there is a subset A* of A with respect to which the method is robust. The

1991 Mathematics Subject Classification. Primary 65N30; Secondary 65N12.Key words and phrases. inf-sup condition, finite element method, mixed method.Both authors were partially supported by NSF Grant #DMS91-20877

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good performance of these methods can now be understood in this context. That is, the

input data chosen for the benchmark problems chosen may just happen to lie in the set

A* rather than A. If such a method is to be of practical use, it becomes important to

characterize the set A* .

In this paper, we shall study a simple one-dimensional mixed finite element method

which will illustrate the ideas mentioned above. The simplicity of the problem will allow

a detailed theoretical analysis and will give rise to illustrative computations. The method

we have chosen to analyze satisfies the patch test criterion, but fails to satisfy the uniform

Babuska-Brezzi condition. We will characterize a set A* of input data which guarantees

good performance of the method. We emphasize that a characterization of A* is simple for

this example. It cannot be readily generalized to other methods satisfying the patch test,

but not satisfying the uniform inf-sup condition. A detailed analysis must be performed

on a method-by-method basis. Strictly for the sake of comparison, we will also study a

mixed finite element method which passes the patch test criterion and satisfies the uniform

inf-sup condition. We will show that although both methods pass the patch test, they have

significantly different robustness properties.

The paper is organized as follows. The second section discusses the general variational

problem which we study. In the third section, we discuss properties of two finite element

methods for the problem. Numerical results are presented in the fourth section. The

implications of our results are summarized in the fifth section. The proofs for all theorems

appear in Section 6. Concluding remarks appear in Section 7.

2. THE INFINITE DIMENSIONAL PROBLEM AND ITS BASIC PROPERTIES

We consider the following one dimensional problem on 1= (0,1):

-s' = I,

u -s=g

u(O) = u(l) = 0

(2.1a)

(2.1b)

(2.1c)

Multiplying the first equation by a function v which vanishes at the endpoints and inte-

grating by parts, J~lsv' dx = J; Iv dx. We next define suitable function spaces in which

2

to cast the problem. Define Hl(I) as the space of functions which, along with their first

derivative, belong to the space of square integrable functions on I, denoted by £2(I). Also,

define the subspace HJ(I) of Hl(I) by

HJ(I) = {vi v E H1(I), v(O) = v(1) = O}.

The weak formulation of the problem is then to find U E HJ(I) , S E £2(I) such that

B((u, s); (v, IT)) = F(v, IT) \;Iv E HJ(I), IT E £2(I)

j'l t j'lwhere B((u,s); (v,lT)) = 0 u'lTdx - Jo slTdx + 0 sv'dx, and

j'l j'lF(v,s) = 0 Ivdx+ 0 glTdx

Define H = HJ(I) X £2(I) with norm

(2.2a)

(2.2b)

(2.2c)

Throughout the rest of the paper, we shall use the HI seminorm since it is equivalent to

the HI norm on HJ(I).

The problem (2.2) has a unique solution and the solution exists for any I E H-1(I),

and 9 E L2(I). The space H-1(I) is defined by

{Ill Iv x ~ CllvIlHl(I) \;Iv E HJ(I) } .

Furthermore,

where C does not depend on f and g. In the subsequent discussion, C will denote a generic

constant with differing values depending on context. The problem (2.2) is well defined for

the input data set A = {(J,g)IIIIIIH-l(I) < 00, IIgll£2(I) < oo}.

Assuming that I and g are smooth, the solution then satisfies

" I '-u = +g

U(O) = u(l)

3

(2.3a)

(2.3b)

From (2.3) we see that if f E Ct(I) and g E Ct+1(I), t ~ 0, then U E Ct+2(I) and

u' E Ct+l(I). Denoting A; = {(J,g)1f E Ct(I),g E Ct+l(I)}, we readily have A; c A,

t ~ O.

We have the following theorem due to Babuska[4].

Theorem 2.1. Suppose:

(1) HI and H2 are two real Hilbert spaces with scalar' product (-, ')H, and (', ')H'2'

respectively.

(2) B( u, v) is a bdinear form on HI x H2, 'lL E HI; v E H2 such that~........ '"'-' ........

inf sup IB(1L,v)I~C2>0,-::EHI vEH2 ~ ~

1I-::lIfl1 =1 1I~llfl'2::;l

(2.4a)

(2.4b)

sup IB(u, v)1 > 0,-::#0 ~ ~

v i- 0, where Cl, C2 < 00.~ (2.4c)

(3) f E H~, i.e. f is a linear functional on H2.

Then there exists a unique element Uo E HI such thatrv

(2.5a)

(2.5b)

It is not difficult to show that Problem (2.2) satisfies all the hypothesis of Theorem 2.1.

3. FINITE ELEMENT METHOD

We seek independent approximations to the u and s variables using the finite element

method. The uniform mesh is given by

o = Xo < Xl < ... < Xn = 1, Xi = ih, h = lin,

with n assumed to be even only for simplicity. Denote the subinterval (x j ,x j+1), j

0,1, ... , n - 1, by I'] and let pt(I']) be the set of polynomials of degree t on I']. Next we

consider the following finite dimensional subspaces of HJ(I) and £2(I).

4

Let

We writen-l

Vh = L ajePj E v,tj=l

n

lTh = L lTjePj E Shj=O

n

a-h = L a-jXj E Sh'j=1

(3Aa)

(3Ab)

(3.4c)

where ePj is the usual linear "hat'; basis function and Xj is the characteristic function on

I~tJ .

We then have

n

IIa-hlli2(I) = h La-j, andj=1

n-l

KllIlThlli2(I) ::; hL lTj ::; K21IlThlli2(I)j=l

with 0 < Kl < K2 < 00 independent of lTh and h.· .

We next define two different finite element methods.

(3.5a)

(3.5b)

(3.5c)

Method A. With the finite element spaces as defined above, find Uh E Vh and Sh E Sh

such that

(3.6a)

where Band F are as in (2.2a) and (2.2c). Analogous to the infinite dimensional case, we- -

define Hh = v,t X Sh'

5

Method B. For this method, we seek Uh E v,t and Sh E Sh such that

(3.6b)

with Band F as given in (2.2a) and (2.2c). We define Hh = \lit X Sh' Note that in Method

B, Sh is used instead of Sh for the second variable.

Theorem 2.1 can be applied to conclude that both mcthods yicld unique solut.ions. It is

important to note that the mere solvability of the discrete system does not say anything

about how well the discrete solution approximates the actual solution. Clearly, each value

of h gives rise to a different Hilbert space, and so the inf-sup constant does not necessarily

have to be the same for each value of h. We shall need the following theorem due t.o

Babuska[4] to address these points precisely.

Theorem 3.1. If

(i) The hypothesis of Theorem 2.1 hold,

(ii) Ml and M2 are given subspaces of HI and H2 respectively such that

inf sup IB(u,v)l2:: d(Ml,M2) d(Ml,M2) > 0,~EMI vEM2 ~ ~

"~"Ml =1 il.0IM2~1

and for every v E M2'

sup IB(u, v)1 > 0, v =1= O.uEM1 ~ ~ ~~

(iii) For given f E H~, let 3:;0 E HI denote the unique element such that

(3.7)

holds for every v E H2 (the existence of such a 3:;0 is assured by Theorem 2.1),

(iv) There exists w E Ml such that

6

(v) There exists it E Ml, Ml is the closure of Ml in HI, such that~o

B(it , v) = f(v),"'0 I'V ,.....,

for all v E M2.~

Then

(3.8)

Here, Ml = 11/12= Hh for Method (A) and 1111 = 1112 = I-h for Method (B). Since

t.hese spaces are finite-dimensional, they are already closed subspaces. The w in the above

theorem refers to an approximation in the subspace whose error bound is already known.

Hence, the error of the computed solution is given in terms of the error of a known approx-

imation and the inf-sup constant. The following theorem gives estimates for the inf-sup

constants for both methods. The proof is given Section 6.

Theorem 3.2. With Hh and Hh corresponding to Method A and Method B, respectively,

we have

(3.9a)

where I is independent of I), and

(3.9b)

where 0 < C1 < C2 < 00 are independent of h.

We note that the patch test applied by itself cannot discern the major difference between

the two inf-sup constants, namely that one is independent of h and the other is dependent- .

on h. For Method (A), we have the following convergence result....--.

Theorem 3.3. Let (1, g) E A and (Uo, so) satisfy 2.2(a)-(c). Then, for Method A

I!(Uh, Sh) - (uo, so)IIH :::; C inf _ II(u, s) - (uo, so)IIH -+ 0 as h -+ O.(u,s)EHh

(3.10)

This theorem is an easy consequence of the fact that any Uo E HJ(I) and So E £2(I) can

be approximated well in II·IIH by functions from v,~and Sh as h -+ o. Thus, (3.10) follows

7

(3.11)

(3.12)

I

I

The prooi in Arnold, Babuska, and Osborn[5] is of an abstract nature and is not con-

structive. Fo.r our cxample, various (f,g) E A for which (3.11) holds can be constructed.

Vie will shoJ a concrete examplc of such input data in the section on numerical results.

While Me~llOd (B) does not perform well for all input data in A, it is possible, at least.I

for this exarriple, to characterize a set A* for which the method performs well. In SectionI

6, we shall prve the following theorem.

Theorem 3t. Let (J,g) E A~, t ~ 2 and let (Uh, Sh) denote the finite element solution

from Method .(B). Then

I

I, I

where II(J,g)llt = max (IIfllet(I), IIgliet-l(I)), and C is independent of (J,g).

We observe that

inf II(u,s) - (uo,so)IIH ~ Chll(J,g)IIt,uEVh

SESh or SESh

(3.13)

and so both methods perform equally and optimally when f and 9 are sufficiently smooth.

Remark 3.1. For sufficiently smooth input data, Method (B) performs better than Method

(A) in the sen'se that for t ~ 2,,

(3.14)I

I

whereas for Method (A), we have only first order convergence for s as given below:I

(3.15)

8

This point is a consequence of the proof for Theorem 3.5, and will not be elaborated• __ u_

n•

further.

The sufficient characterization of A* that is given here for the optimal performance of

Method (B) is simple. The characterization can still be somewhat. generalized, but will

not be done in this paper. We emphasize again that a characterization based only on the

regularit.y of the input. data does not necessarily generalize to any met.hod satis(ying the

patch test but does not satisfying the uniform Babuska-Brezzi condition.

Relating the performance of Method(B) to the nature of its input data gives a practical

guideline with which we can judge the applicability of this particular method. Because of

the correspondence between the solution and the input data, we can equivalently relate

the performance of the method to the properties of the solution. Therefore, instead of

the assumption (f,g) E A (respectively, (J,g) E An, we can just as well assume that

(uo, so) E H (respectively, 'Uo E C4(I), So E C3(I)).

4. NUMERICAL EXAMPLES

To illustrate the type of results obtained when Methods (A) and (B) are implemented

for problem (2.2), with 9 = 0, we assumed that the exact solution for the u variable is given

by u = Ix - 0.510- 0.50

, where 0:' was allowed to assume a variety of values as long as fwas square i~tegrable, i.e. 0:' > 1.5. Exact integration was used to compute the right hand

side and the various errors. Therefore we can be assured that the observed behavior of the

errors stem only from the methods themselves. We compute relevant error quantities for

the two methods and summarize the results in the following figures.

4.1 Rate of convergence for Uh. Figure 1gives the rate of convergence for Uh for various

values of 0:'. Method (A) gives optimal results for all values of 0:' > 3/2, as predicted by

Theorem 3.3. For Method (B) the rate is of order h for a ;::: 3, that is, when Uo E C3(I).

9

0.8-x- MethodA

-0- Method B

0.4

0.2

o1 1.5 2.5

alpha3 3.5 4 4.5

Figure 1 Rate of convergence of u vs. CY

4.2 Ratio of actual error to interpolation error. In order to get an idea of the size

of the constkt appearing in the approximatio'n result of Theorem 3.1, we plot the ratioI

of (lluo - uhll~-[l(I) + Iiso - shlli2(I))1/2 to (Iluo - Uintllt-l(I) + Iiso - Sintlli2(I))1/2. The

results are shbwn in Figure 2. Here, Uh is the computed solution and Uint is the piecewise

linear interpJlant to Uo; likewise for Sh and Sint. The results are for a fixed mesh with

h = 1/256.

Note that IMethod (A), the one with the uniform inf-sup constant, has a ratio which

remains the same regardless of the value of CY. However, for Method (B), the ratio

does not seem to stay bounded as CY gets smaller, corresponding to a less smooth so-

lution. This I behavior is not surprising since we know that the inf-sup constant is ofI

order h2, and by examining the approximation result in Theorem 3.1, we see that the

ratio may ndt necessarily be bounded for the second method. Direct application ofI

Theorems 3.1, and 3.2 would suggest that the ratio is large for all CY. However, Figure

2 shows that I the ratio is smaller than expected. This will be addressed in Section 5.

10

10'

-1(- Method A

-0- MethodS

.2 JO'e

100.... )(-.'t(.tO<-tOl-.- - -f(-·-·H ->'- - - - -t(.- -·-·-x

1.5 2.5 3.5 4.5

Figure 2 Error ratio VS. a

4.3 Relative error for u vs. a. Since for practical computations, we are mostly in-

terested in the relative error, we next plot the relative error of Uh as a function of a

for a fixed mesh with h = 1/256. Again, Method (A) performs well regardless of the

value of a, while Method (B) deteriorates for a < 2. This is illustrated in Figure 3.

Once more, we see that the performance of Method (B) is related to the smoothness

of the solution. Figure 4 shows an enlarged plot of the relative error when a ~ 2.0.5

0.45

0,4

0,35

0,3-x- MelhodA

025-0- MelhodB

0.2

0,15

0,1

0,05

°1.5 2 2,5alpha

3 3.5 4

Figure 3 Relative error of U VS. a

11

X 10-314

10

6

12

- -.x- _

22 22 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4

alpha

Figure 4 Relative error for ex~ 2

1 5. THE BABUSKA-BREZZI CONDITION

In this section, we elaborate on the basic Theorem 3.1, and explain the relation betweenI

it, Theorem 3.5, and the reported numerical results. We begin by examining the proof of

Tl~eorem 3.1.1Let Uo E HI satisfy

I B(uo, v) = F(v) \;Iv E H2·III

Further, let u~ EMf C HI satisfy

I B(Uh,Vh) = F(Vh) \;Ivh E MI; C H2.

IWe also have [that

. B( uo, Vh) = F( Vh) \;Ivh E MI;.

Let Uh EMf lnd Uh = Uh + z, z EMf. ThenI

B(z, Vh) = B(uo - Uh, Vh) \;Ivh E MI;.

and so

sup IB(z, Vh) I :::; Cilluo - uhllRl'IlvhllH2=1

(5.1)

12

where Cl is given in (2.4a). Now for any z E !I;[tL we have

C111uo - uhliHI 2:: sup IB(z,Vh)l2:: d(Mit,MiL)llzIIHIIlv"IlH2=1

and therefore,

Since Uo - Uh = Un - flh + z, we get

(5.2)

(5.3)

(5.4)

We now analyze this proof to see if it is possible to improve certain estimates under

additional hypotheses. We used two relations which could possibly, but not necessarily, be

pessimistic. Relation (5.2) holds for any z E !I;[tt, but we need it only for z = ih - Uh'

Therefore, we need only estimate

(5.5)

We then have the following estimate instead of (5.3):

(5.6)

In Section 6, we show that for Method B, we have for Uo E C4(I), .

(5.8)

with Uh = ubnt, where ubnt is the interpolant of uo. This is an improvement over the O(h2)

estimate given in (3.9b). The second pessimistic estimate was used in (5.1). Here, we used

(2.4a) which is essentially the Schwarz inequality

(5.9)

13

,

But all that ~e needed was

iIB(uo - Uh, Vh)! :::; K(uo - uh)lluo - uhIIHlllvhl/H2• (5.10)

(5.12)

Noting that I< is a function of Uo - Uh, and assuming that Uo E 04(I), we can arrive atI

the estimate

K(uo - Uh) :::; Ch, (5.11)

where C is ~ollnded independently of h, but can depend on lIo- As before, il" 'c 1I.g"Therefore, inrtead of (5.4), we have the following:

_ ( K(UO-Uh)) _ _Iluh - uhllHl < 1+ d(- _ 1I1h) Iluo - uhllHl :::; CI/7.l - uhllHl·

l Uh Uh, 2 -

More preci ely, we can ask what the status of the inf-sup constant would be if we were

to examine a ]subset of functions in vi, x 5" wbose oscillatory behavior can be controlled

in some fashi~n. Indeed, such a restriction improves the quality of the inf-sup constant, asI

is made preci~e in the following theorem.

I ~

Theorem 5.lI.. Let Vh consist of all functions in Vh with lu~1 :::; K2 on [0,1] and, zn

addition, satifJYing. one of the following conditions:

(i) lu~1 >IKllluhIIHl(I) on (0, h), for h small enough.I

(ii) u~ > f21/uhllHl(I) on (Xi, Xi + h), (Xi - h, Xi) for h small enough, for some i .

(iii) -u~ ?, K31/uhIIHl(I) on (Xi, Xi + h), (Xi - h, Xi) for h small enough, for some i.I

Here, Kj > 0, j = 1,2,3 and are assumed to be independent of h. Then there existsA I

i(Vh x Sh; Vh:x Sh) such that

I

inf sup(Uh,Sh)EVh XSh (Vh,Uh)EVh XSh

(Uh,s~):;eO (v,,,uh):;eO

with Clh S j(ir. x 5,,;Vh x 5h) S C2h, Cl and C2 independent of h_ Hj

HJ(I) x £2(0,11).

The followihg theorems will characterize A* and its resulting properties. They will be

d· hi.prove m t e next sectIOn.I

14

Theorem 5.2. Assume that 9 = 0, f E Ct(I), t ~ 2. Then the solution Uh of Method B

belongs to \liL' The assumption 9 = 0 is made only for simplicity.

Theorem 5.3. Assume that f, 9 are as above and let 11,int E V/L be the interpolant to Uo.

Also let (~:o)int = (SO)int = Sint E Sh be the interpolant of so. Then

6. PROOFS OF THEOREMS 3.2,5.1,5.2,5.3

6.1 Proof of Theorem 3.2.

Proof of Inequality (3.Da): Given (Uh, .'ih) E V/Lx Sh,we choose Vh = 211,h,lTh = -Sh +U~L'

Note that U~L E Sh. Theil

t j'l j'lB((Uh, Sh); (Vh, lTh)) = Jo

U~L(-Sh + u~J dx - 0 Sh( -Sh + u~J dx + 2 0 U~LShdx

=11

u~2dx +11

s~ dx = II (Uh, sh)llk.

from which (3.9a) directly follows.

Proof of Inequality (3.9b) First, we prove the left hand side of this inequality. Given

(Uh, Sh) E Vh x Sh, we choose lTh = a-h + o-h, Vh = Vh + Vh with a-h = -Sh , Vh = Uh. The

quantities o-h and Vh will defined explicitly in what follows.

We first have

(6.1)

(6.2)

15

where11.

a-2) = b I)-l)n-i¢ii=O

a-~3) = C

Since a-k2)(i(L) = b(_1)(n-i), we easily see t.hat

/

.1,_(2) _(3)

'/Lh((Th + lTh ) dx = 0, 0

{or any b an~ c. With (,i = (Li - o.i-l, we define a-~1) E S" by

(6.3)

(6.4)

11.

_(1) _ -(1)(.,/) _? ~ (_l)j+i-lc,(Th,i - (Th t L - ~ L <'J'

j=1+1

i = 0, 1, .. " n - 1 (u.5)

and a-(1) = ~(1)(1) = O.h,n I h

We next estimate the norm of a-h. By the Schwarz inequality,

11. 1~(1) ~( ,)2)1/2_

IlTh,i I :::; 2(L.. (,J hl/2 'j=l

and· therefor

i = 0,1, ... , n,

using (3.5a).1 Furthermore,

1 n (~(l)+A(l) ) n 1I ~(l)d _ ~c. lTh,i lTh,(i-l) _ ~c2-h r 12d

UhlTh x - L..<"z 2 - L..<"i - io Uh X.o i=l i=l 0

We next Jlect Vk E v" depending on ih so that

(6.6)

(6.7)

(6.9)

We also need to define band c in (6.3) and (6.4). To this end, Lemma 6.1 summarizes

essential pr01erties stemming from (6.9) and will be proven after the proof of Theorem 3.2

is completed.

16

Lemma 6.1. There exist band c in (6.3) and (6.4) with

Also, there exists Vh E Vh satisfying (6.9) with

where C is independent of h.

From (6.3),(6.4),(6.6) , and Lemma 6.1, we have

(6.10)

We defined Vh = Vh +Vh = Uh +Vh. Substituting into the bilinear form, and using (6.7),

we have

Next, we have the following estimates for the norms of lTh and Vh:

and so

Using (6.12) and (6.13), we have

17

(6.12)

(6.13)

thus proving[ the left hand side inequality (3.9b).

Proof of t{e right hand side of (3.9b):

Let f(x) C=(I) vanishing on (0, a] and (I-a, 1], with 2a < 1, and f(x) > 0 elsewhere.

Let fih E V/L e defined as

(6.14)

For h small elnough,

(6.15 )

Further let. sI1 E Sh such t.hatI

VYethen have~ for sufficiently small h,

t h2(-1)iJo sh4>i dx = - n (4j'(Xi) - j'(xi+d - j'(xi-d)

= _h2( -l)i[J' (Xi) + O(h)], i = 1,... ,n

and

(6.IG)

(6.17)

Therefore,

(6.19)

recalling the f~ct that are no contributions from both ends of the interval. Letting lTh =

18

Denoten.

Vh = L di<pi, do = dn = O.i=O

Then,] n

In ,- h ~ di - di-1 ( )VhSh dx = - 6 Sh,i + Sh,i-l

.0 2 , h1.=1

where Xi-l < f.i < :ri'

Therefore,

(6.21)

(6.22)

This proves the right hand side of inequality (3.9b).

Proof of Lemma 6.1:

Denoting Vh = L~::'llvi<!Ji, (6.9) leads to a system of equations for Vi, i = 1,2, ... , n - L

and the constants band c. This can be written in the form

A~ =µ,~ ~ (6.23)

where A is the matrix {aij}, with i,j = 1,... ,n+ 1, ~ = (Vl"",Vn-l,b,c) and I!:,-

(µo, ... , µn). From (6.9), we get

_ h (2 ~(1) + ~(1»)µo - '6 lTh,O lTh,l

j = 1, ... ,n - 1

(6.24)

19

For the matl1ix A = {O'.ij}, we have,

12' i=l, ... ,n-l

1O'.i,i-2 = 2' i = 3, . , , ,11. - 1

hO'.i,1/. = 3' for i even, i = 2, ... ,11.- 1

11,0'.7" 7/ = - - for i odd. i = 2. ' .. , n - 1, "3 '

Cti,n+l = -11, for i = 2, ... , n

h0'.1,1/. - 6

hO'.n,n - 3

O'.n,n+l = -h

hO'.n+l,n+l - 2

(6.25)

and all other cbefficients etij are zero. By elimination, A can be transformed to the upper

20

triangular matrix A = {iiij} and the right hand side {µj}. Simple computations yield

1iii,i = 2' i = 1, ... ,71, - 1

n -1i = 1, ... , 2

11.ii2,n - 3 (6.26)

11.iil,n+l - 2

71,i = 2, ... ,?

11.ii2i+l,n+l = -11. + ii2i-l,n+l = -2 - ill., !2:-1i = 1, ... , 2

ii2i,n+l = -h + ii2i-2,n = -hi,

2- - -1 - -lXn+l,n+l - 71,

71,i= 2, ... , 2

and all other coefficients are zero. Recall that 11. = 1/71, and that 71, is even. We also have

Hence, for band e we get the estimates

lbl :S Cllo-~l)II£2 (l)

lei :S clla{l) 11£2(l)

(6.26),(6.27), and (6.28) lead to the statement of Lemma 6.1.

(6.27)

(6.28)

6.2 Proof of Theorem 5.1. First we prove the lower estimate of ,(h) for case (i). The

proof will be analogous to the proof of Theorem 3.2. Let o-h be given as in (6.2) but set

~(l) - f,(Yh,O - ,

~(1) - 0(Yh,i - ,

21

i = 1, ... ,71, (6.29)

where f;. = IIl-th IIHI (I)' This gives

From our assumption,

As in (6.8), o/e choose,

and construcl Vh E Vh so that (6.9) holds with lTh given in (6.31).

Using the lsame proof as for Lemma 6.1, we get the estimates,

lui ::; Clo-~~6Ih

lei ::; Clo-~~6Ih

IlvhllHl(I) ::; Clo-~~61 ::; Cf;. = ClluhllHl(I)

and as in thel proof of Theorem 3.2, we have

Since

IllThlli2(I) ::; C[lIshlli2(I} + hlluhllk1(I)]'

Ilvhllk1(1) ::; C[lIshlli2(I} + IIuhlli2(I)], we have,

II(Uh, sh)IIH > C.II(Vh, lTh)IIH -

(6.30)

(6.31)

(6.32)

(6.33)

Hence, the Imyer estimate for case (i) is proven.

If conditiorl (ii) holds, we set o-~~l = lIuhIlHl(I) and the other nodal values to zero. In

Case (iii) we 1et o-i~l~ -lfuhIlH'(I) and proceed as before.

We next prove the second part of Theorem 5.1, namely the upper estimate for 'Y( h). The

proof is analolous to the one for Theorem 3.2. Let f(x) E COO [0, 1] such that f(x) 2:: 0,

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1f(O) = 1, and 1'(0) = 0 and f(x) vanishes on [1- a, 1], 0 < a < -. We define Uh E Vit as

2follows:

We set Uh,O = O. We have lu~Jx)1 ~ I<lIuhIlHl(J), I< > 0, for 0 < x < h. Also,

Next, let Sh E Sh such that

For sufficiently small h, we have

'i = 1,_ .. ,11

Furthermore,

i= 2, ... ,n

Therefore,

and

With lTh = 2::7=0 lTh,icPi, we have

/11

ShlTh dx - 11

UhlTh dxl ::; IlTh,olh + IlTh,llh + Ch21IlThIlU(I)

::; ChlilThIlU(I)

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Since Sh,O =lh.f'(O) = 0, we get the following result, as in the proof of Theorem 3.2.

Hence, we h~ve constructed a (Uh, Sh) E Vj~x Sh such that

This leads to the upper estimate of Th for case (i). The cases (i) and (iii) arc proven

analogously ~'Sing f = 1 on (:Ci - 11., :Ci + h).

I6.3 Proof of Theorem 5.2. Let Uh = Uh + {Lh, 'Uh = Uint and Sh = '~h + Sh, Sh = Sint'

Let Sh = 2:::10 Sh,i<Pio We have

i=l, ... ,n-l.

Then {Sh,J s~tisfies the system of equations

i= 1,. 0" n - 1, and

(6.34)

Then S = s(tl) + S(2) + S(3) whereh h h h'

, s~l}= O(h2),n

~(2) _ b"(_l)n-iA.. ~(3) _S h - ~ 'f't, S h - C

i=OI

Let Uk = ~~:II v.p, Then the vector (VI, ... , Vn-I, b, c) satisfies the system of linear

equations (6.2~) with Iµjl = O(h3), j = 0, ... ,n. As in the proof of Lemma 6.1, we have

I' Ivd :S O(h2).

Hence, for ~ufficiently small h, there exists Xi for which either (ii) or (iii) is satisfied.

Remark 603.1'1 The validity of the estimate (3.13) is evident from the proof of Theorem

5.2

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6.4 Proof of Theorem 5.3. Denoting Vh = '£::11ad)i,

11.' (U~ - U:n, - (so - sin,))O",,1 S hI: lO"h,iIO(h') + (10"",01 + 1O"",nI)0(h2)1,=1

:S O(h2)lllThIIU(I),

from which the statement of Theorem 5.3 follows.

7. CONCLUSION

Through the analysis of a simple example, we have seen that simplified conditions used

in place of the Babuska-Brezzi condition do not fully discern the robustness properties of

the method. For the method we have analyzed, a ·sole application of the engineering patch

test does not reveal the dependency of the inf-sup constant on the mesh size h. It also

does not give information about the types of input data that would yield optimal results.

On the other hand, a direct application of the standard mathematical theory involving the

Babuska-Brezzi condition does not fully account for the good results seen in some finite

element methods used in engineering practice. With a more refined analysis, we were able

to characterize a set of input data, referred to as A*, which yield optimal convergence

results for the finite element solution. For this simple example, the set A* depends on the

smoothness of the exact solution. We also showed that the inf-sup constant's dependency

on h improved from O(h2) to O(h) when a "better behaved" subset of the finite element

space was considered. Hence, we were able to clarify contradictions which arose between

the mathematical theory and actual practice.

25

REFERENCES

[11 R.L. TaJor, O.C. Zienkiewicz, J.C. Simo, and A. Chan, The patch test - a condition

for assessing I FEM convergence, IJNME, 22 (1986) 39-62,

[2] O.C. ZietkieWiCZ, S. Qu, RL. Taylor, and S. Nakazawa, The patch test for mixed

formulations IJNME, 23 (1986) 1873-1883.

[3] O.C. Zie1kiewicz and R.L. Taylor, The patch test - necessary and sufficient conditions

for convergeJ'ce, TICAM Summary Notes, University of Texas, 1993.

[4] A.K. Aziz and 1. Babuska, The Mathematical Foundations of the Finite Element IVfethod

with Applica ion to Partial Differential Equations (Academic Press Inc., New York, 1972).

[5) D.N. Arntld, 1. Babllska, m;d J. Osborn, Finite element methods: principles for their

selection, Corp. Meth.. in Appl. Mech. and Engrg., 45 (1984) 57-96.

TEXAS 1NSTlTUTE FOR COMPUTATIONAL AND ApPLIED MATHEMATICS, TAY 2.400, UNIVERSITY OF

TEXAS AT AUStIN, AUSTIN, TEXAS 78712

DEPARTMENjr OF MATHEMATICS, SAINT PETER'S COLLEGE, JERSEY CITY, NEW JERSEY, 07306E-mail addTtss: narasimhan_r@spcunb.spc.edu

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