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UARI Research Report No. 132AFOSR Scientific ReportProject THEMIS AFOSR-TR-73-01ll

CONVERGENCE OF MIXED F INITJo:ELEMENT APPROXIMATIONS

OF A CLASS OF LINEAR BOUNDARY-VALUE PROBLEMS

hy

J. N. Reddy and J. T. aden

Research Sponsored by Air Force Office of Scientific ResearchOffice of Aerospace Research, United States Air Force

Contract No. F44620-69-C-0124

The University of Alabama in HuntsvilleSchool of Graduate Studies and Research

Research InstituteHuntsville, Alabama

Octoher 1972

Approved for public release; dIstribution unlimited.

CONVERGENCE OF MIXED FINITE ELEMENT APPROXIMATIONS OF A CLASS

OF LINEAR BOUNDARY-VALUE PROBLEMS

t.T. N. Redcly* ilnd .1. T. Ocl(!11

I. INTRODUCTION

In the pres('nt paper, w(· prove convt'rgence and der i VI' ~elH'ra1

properties of mixed finite-element models of a general class of linear

boundary-value problems of the type

Au + ku + f = 0,(1.1)

when, \1 = u(~) i9 fl function d(·fjIH·d on II hounclpd rl'glon~"i of E", ~~ is

&-,

H(II - g) = 0 on 2W.1 B*(TII - S) = 0 on OSS

the smooth houndllry of b~, ~ is n point in R, !\ Is ali near factorable

op~rator, k is II pos1.tiv(· consl"Ant, and IIand 1\>'" an, opt,rators dpscribing

mixed boundary conditions on alii. The operator A is assumed to

have the following properties:

(i) A is factorable, in iliesense that

A T*T (1.2)

where T is a linear operator from a Hilbert space U into a space 'JJ' and

T* is i tB formal lldjoint.

(i1) A is coersive; i.e. If (.,.) denotes the inner product 1n'/..{,

then there exists a positive constant c such that

*tSenior Research AssistantProfessor of Enginee ring Mechani cs

(Au,u) + k(u,u) ~ c· (u,u)

2

(1.3)

Equations of type (l.l) occur frequently in structural mechanics.

To cite a few examples,

, 1. Stretched cord with elastic support:

d·u- II dx2 + ku = f (x) (1.4a)

Here T = d/dx, T* = -d/dx,H is the horizontal component of tension in the

cord, and k is the measure of spring constant of the support.

2. Beam resting on elastic foundation and subjected a transverse

load f (x) :

f(x) (1. 4b)

Here T = d2/dx2, T*= d2/dx2, E(x)I(x) is the flexural rigidity of the

beam, k is the foundation modulus, and u is the transverse diaplacement.

3. Membrane on elastic foundation:

ifc!u + ku = f(x,y)

Here T = grad, T* = -div, and k is the modulus of the support.

4. Thin elastic plate on an elastic foundation:

lJ4u + ku = f(x,y)

(1. 4c)

In this case T = T*

modulus, etc.

V2 = ?? /?Jx2 + 02 /oy2, k is again the foundation

,...For years the use of either 'displacement" or "equilibrium" finite-

element models dominated all applications of the method to problems in

structural mechanics. Both types of models precipitate from well-

established extremum principles; displacement models are generated from

certain minimum principles (e.g., the principle of minimum potential

,

3

energy) and equilibrium models are derived from complementary maximum

variational principles, or their equivalent Galerkin integrals. In

either type of approximation, it is now a routine task to establish con-

vergence criteria and to develop sharp estimates of the rates-of-convergence

in various energy norms.

In the mid 1960's, however, use of so-called mixed finite-element

models for plate bending were proposed by Herrmann [lJ. These involved

the simultaneous approximation of two or more dependent variables and were

based on stationary rather than extremum variational principles. Dunham

and Pister [2] used the Hel1inger-Reissner principle to develop mixed

finite-element models of plate bending and of plane elasticity problems

with surprisingly good results. parallel to the work on mixed models was

the development of the closely related hybrid models by Pian and his

associates (e.g. [3]), and Wunderlich [4J exploited the idea of mixed

models in a finite-element analysis of nonlinear shell behavior. More

recently, Oden [5] has discussed some generalizations of the ideas of

mixed models. In all of these studies, results of numerical experiments

suggest that mixed models can be dev~loped which not only converge very

rapidly but also yield higher accuracies for certain quantities (e.g.

stresses) than the corresponding displacement-type model.

f)~spite wide-spread use of mixed finite-el~ment models and the

advantages they appear to offer, their mathematical properties are not as

well understood as those of th~ displacement or equilibrium models. In

constructing mixed models, the convenient extremum properties of the usual

displacement or equilibrium approximations are lost, and the ppsitive-

definite character of the associated functionals is, of course, non-

existent. As a result, there does not appear to be available a study of

general properties of mixed finite-element models or proofs of convengence

,

of Stich models for general linear boundary-value problems. We attempt to

examine these issues in the present paper.

In the sections following this introduction, we hriefly outline Borne

mathematical preliminaries for formal development. The notion of pro-

jections and approximation of operators hy projections are sumnwrized, Rnd

finite-e lement approxim;]t Ions of mixt·d models arc discussed I n Section 2.

In SectIon J, we present mixed variation;]l principles for hrnlndary value

problems of the type (1.1). We also show the n'latlonship of Ritz-type

npproximations obtai.ned using these v!lriational principles to tl1(.' projl'c-

tions discussed earlier. In Section 4, we present a convergence proof

set of II-linearly independent e It:ments in 'lJ' .•...

for mixed finite-element approximations, and give a posteriori error

estimates. Finally, we present a simple numerical example to clarify the

ideas.

II. SOME MATHEMATICAL PRELIMINARIES

Tn this section, Wt' rl:view hrlefly certain mathematical properties of

mixed fIllite-c!<'menl: approxinwtions lllllt arc eHentlal for our study. For

a more dct;~j]cd aCCollllt of the llll'l)ry of finlle-l'leJllents devc]opl~d from

the notion of projecti.on opl·ralors. ont' can cOllsul1 [6J or, for SPCCllll

attention to mixed probll'llls ['l].

2.l Projection of Operntors. Consider two linear vector spaces, U and '?c.defined over the saine field, and let (cp }. (Ci = 1, ... ,G) denote a set of

Ci

G-lincarly independent l'll~ml'nLs ill't.{, [lI1d C~{~},(6 = 1,2, .. ,II) denote a

The sets (cp } and (wlJ.} defi neCi -

G-dimensional subspace Tl1.Q C U and H-dimensional subspace KH C t respec-

tively. The Gram matricl:s associated with the subspaces Tl1.Q andJ{'1-l are of-the form

(2. I)

where (. ,.} and [. ,.J ar~ l.h~ inner products in U and 'lI rt~Sp~cti.vely.

Since [, } and [w6} are linearly independent, the matrices C Q anda - ap

H6r

are nonsingular, and we can compute directly biorthogonal hases

"

r116r~ (2.2)

FromHere GaS and 1',.,1' ;Ire the i Ilverses of GaS lind Hl.\l.~rr'spectlvely.

(2.1) ;lI1d (2.2), it is clear that Wl' have the hiorthogonality conditions

(2.3)

Note that without additional information there is no relation he tween the

spaces U and 'lJ', and the biorthogonal hases in TT\.Q and JrH are completely- -independent of each other.

We now define orthogonal projection operators n: '/..( -+ T11o, and

p: t -+ ~H in the followin~ senSl': j r II is an arbitrary elf!mcnt In'/..( and

v is an arhitrary l~h~mL'nt: in 'lit thl' projection ~ of 11 into TT\.1l and the- -projection v of '! into JfH are of thl' form

n(ll)

G

II = L ;]a'a and rev)Ci

(2.4)

where aa and b 6 are given by

(2.5)

•

Now let U be a Hilbert space consisting of functions u defined on a com-

pac t, convex subse t R of En wi til a smooth boundary ow., and 1(' t T he A

hounded lilwar operator mapping'/..( into another IIi Iberl space 'lJ'. Then the

operator T*:! -+ 't{ is the forma 1 adj oint of T if it sa Us fi es tlH~ gel1l'rall zed

Green's formula

(2.6)

6

Here the last term consists of boundary terms; H is a linear operator

which depends on T, and has an adjoint B* in the sense that

(2. 7)

"We consider the caseS in which mIl C D(T), tIll: domain of T, and JfH C D(T*),

the domain of T*. In genera l, T (mG) is not a subspace of J[H' and T* (~ )

is not a subspace of lTlQ• If D(T) = U or if R(T) = U, then T Is said to he

self adjoint, and U = !.We can approximate the operators T and T* by projecting T(lTlQ' into

~ and T*(JfH) into lTlQ• This projection process leads to a number of

rectangular matrices of which the foIling are of interest in the context

of the paper.

,

L .APT (lTl!)) : P(TCPa) = ma ~

6

nT)"'(~) : 6 L A an(T*w ) = n cP~ - .aa

(2.R)

(2.9)

Here P: !-t~iand 11: 'U, -tlTlQ Rn! projection operators defflll'e1 by til('

bases CPaand ~ (see (2.4)J, and

.6-ma

6-(Tcp ,w J;a-f'ln .cr (2. 10)

•

Similar projections can h(~ obtai ned for Band B*.

2.2 Approximation of Operators by Projections. The ielea of projections

of linear operators cnn he used 88 a hasi s for 11 theory of mlx(~d approxi-

mations. Though there t'xists a nurnher of possible projections for a glv('n

operator, we cite here the four important projections indicated in [7]

which appear to be the most natural. It should be noted that the dual

and mixed projections arc q\Jite different from that given in [5] .

T*TII + kll + f(x) ::0 0 in R, k I- 0

7

B(u-g) = 0 on oR1 B* (1'\1-5) = 0 on ~ (2.11)

where II, f €U, T is a linear operator fromU l.nto 'JL, T* iB itB formal

adjoint as defined in (2.6), k is a positive constant, g Bnd 5 are preB-

crihed functions on the hOllndaries Obi 1 and oRC! respectively (ow,l (}1 oRC! =

OR), and Band B* satisfy (2.7). We can split the problem (2.11) into

the familiar canonical form by defining

Tu == v in hi B (g-u) = 0 on ~'1 (2.12)

T*v -+ k\l + f = 0 in R, R*(Tu-S) = 0 on 0w'2 (2.13)

One shO\lld note that the eqn. (2.13) iB not completely defi nee! in terms of

v. To do so, we operate on (2.13) hy the operator T to get

TT';cv + kv -f Tf = 0 in R, (l<-lll(T*~'1 f + I<g)) c () on ~1 (2. l4)

where it is assuml·d that Tf l'xists. We sl'ck wcak Boll1tlons of \2. tl-2.14)

in subspaces lTt(J' tTtGtl.JJ{; .. , andJ{'Hhy liSe of projections discuss('d at tlw,.." ~

beginning of the section.

2.2.1 The Primal Projection. Wl' seck an approximate solution n(lI) = \J

of primal prohlem (2.11), and req\Jin' thAt thl' projection of thc n:sidlllli

into lTlG v'lI1ish. That is

TI (T*T (11\J ) I I< n(\I) ~ f) = 0 i 11 6t

n(B*«'TTlll) - S)) = 0 on ~ (2.15)

This is equivalent to tile conventional Ritz-Galerkin approximation. We

satisfy the boundary condition on ~l hy proper selection of the hasis

8

and slJbstil:\Iling Lnto (2.15) Wl' get

We assume thal the projection 1I of II is of the form

in R.o

(B'>'«T( L aO/cpCf) - S) J <Pa}OR2

cpa = 0Cf

Since cpa are linearly independent, it follows that

2: aCf(T~\-'fCPCf,CPa} + k IaO'(cpCf,cpe} + (f,cpe}Cf • Cf

o in R. (2. 16)

L aCf (B*Tcp Cf' <re} arl.:a = lI~*s,cpa) ~2

Ci

(2.17)

Using the defInition (2.6) clOd (2.17), the first term in (2.16) can be

written as

LaCY(T*TCPCi,CPe} = L aCi(TCPCi,TCPaJ - L aCY(B*TCPCi'CPa}~l- (B*S'CPa}~Ci CY CY

This leads to a system of linear nlgehraic equations

GL aCY(Nae + kCae) + fa = 0

a(8 1,2, ... ,G) (2. I R)

where

(2.l9)

Solution of (2.l8) determines the coefficients

to the approximate solution u = L aCYcpO'.

0/

0'a , which in turn leads

9

2.2.2. Dual Projection. As the name itself indicates, this is dual to

the primal projection. "ere we seek the approximate solution n(~) = Y..

of the dual problem (2.1/.) requiring that the projection of the rcsidllal

into cJtH vanish.

P(TT*P(y) + kP(::) + Tf) o in bt

(2.20)

P(B(f + kg) + BT*P(y) :c 0 on aot1II

Again, we assume that y.. is of the form y... = 2: b b,5E..b.. Substituting

(2.20), and recalling that wr are linearly i~dependent, we arrive

into

at

another set of linear algebraic equationsH

Lb.

where

o (r,., l, 2 , ... , H) (2.21)

f' b. rMM' == (T*':E.b., T*~ ) ·1 (II (T*':E. ), ':E. } a~~' b. r

II =[w,w]- -rg (2.22)

- ~ b.Solution of (2.21) leads to the approximate solution y. = ~ ':E. .

b.2.2.3 Primal-Dual Projection. This, together with dual-primal proiection,

to he discussed suhseqllt'ntly, leads to the mixed fonnulation of hlHlOdary

value problem (2. I l). Assuming the!! T*v E: w'(n), WL' seek approximatl'

solutions II, V of (2.13) such that

n(T*l' (y) + k (n\!) + f) ,., 0 I n ~

n{B*{p(y) - S)) = 0 on CR2 (2.23)

Using the similar procedure we used in primal and dual projections we

arrive at a set of equations

where

Ib~M~a + kI~ a

o 1,2, ... ,G)

10

(2.24)

(2.25)

..

and Caa and fa are [IS defined in (2.19). Since, In gl'IH:'ral, TTlil and

cJ!:H ilre of djfferent dimensions, M~S i9 il rectangular matrix; and sinc!'

(2.2/1) involves (C -t II) unknowns with only G equations, no unique solution

to (2.2lJ) exists. \.Je need II more eqllations with the sam(' varl.ah1es.

These equations are provided hy tIle dlllll-primal projection.

2.2./, Dual-Primal Projection. Again, we assume Tu E ~(P), and requi.re

that the projection of the resid\lal of (2.12) vanish in ~.'"

P (T (TIU ) - p ('0) = 0 in R(2.26)

P(fi(g - (ml))) = 0 in ~l

This leads to

where

<l~' ra o (2.27)

r r(cpa,T*I.!:l. } ·1 [J\CPI'¥'~ JCbY.

2

I' rg' = [Bg,~ JOSt

1

(2.28)

Equation (2.28) involves «(;.j- II) unknowns alld only II <,qlJations are

availahle to determinl' hoth aei [ll1d h~. However, (2.2lJ) in ctlnjlJnctlon

with (2.27), leads to il deu,nnin[ltc system for tIll' [Jpproximat(~ solutions

U [lnd v. From (2.27) Wl' 0l>l8 in

(2.29)

suhstituting l·qn. (2.29) Into (2.2/1) gives

1]

Substituting cqn. (2.29) into (2.24) gives

~- a~(Kaa + kGaa)a + fa = 0a

where

(2 30)

(2 . 3\ )

8. j'In most cases, M c is till' transpose of N' , a fact not bbviolls from the

'1) a

dev~lopment. Then KaS is syrrunetri c. I':quat Ion (2.30) determi nes the

coefficient [ja and hence gives the approximate solution 7;. Hy suhstitu-

tion into (2.29) we determine b~l which defines the approximate soltltion

v.

2.3 Finite Element Approximations by Eroj~ctions. The distinct property

of finite element mL·thod \olhich makl's it so powerful and convenient is

that the essential feat\lres of an approximation can he described locally

for typical disconlll:ctNI elements. lIer~ we give thE' resulting finite-

element models for primal, dual, llnd mixed projections without going i.nto

the details (see [bJ).

2.3.1 Primal Finite-Element Model.

projections of the form

µ.IIL're Wl' seek coefficients A among

U(x) Lµ.

~(~",(J (x)L µ.'l'N -

N

(2. 32)

which satisfy

L8

= 0 (2.33)

12

(e)Here ON is the boolean transformation matrix,µ.

1 if node xµ' of the finite element model R of 6i

~ + 06i is coincident with node xN or the finite element:~ "

o otherwise

and 1jI~e) arc the local interpolation functions having the properties

(2.35 )

o~, ~~ being the local coordint:es

Due to the particular choice of basis functions, we note that

and fCJ LL

(e) N

(2.36 )

..where k~~ ,Ind f~.,) <Ire the local izations of Ka9 and f~e):

k(e)NI~

(2.37)

2.3.2 Dual Finite-I~Lement Models. Similarly, we seek coefficients B6

among the projections of the form

which sat is ry

where

L p6rl~6+gl' = 0

6

(2. 38)

(2.39)

where

LiI' ILL(")/)" ("h'I' = (2 pNI~~llN (e) M

(l') N M

I' "qj'~~~ = L L ~N g~e)

(l') N

13

(2.40)

,(2.41)

(r)6.and ON is boolean transformation matrix, and j!Je) are the local interpola-

tion functions, satisfying conditions similar to (2.35).

2.3.3 Mixed Finite-Element Models. Finite element approximation of

(2.12) and (2.13) leads Lo mixed model of the form (2.30). Conslder rn.Q

6.to he a subspace of U ;]nd ~i be ;1 subspace of ! spilJ1ned by q>Q' and ~ '

respeclivldy. We sl'ck cOl'J'ficients AA. lind B{)" ilInong projections of the form

u(~) = L A 'A q>A. (~)A.

and sa t is fy

(2.42)

and

L>6.M~a + k IAI'YCQ'6 + fS{)" (y

o (2.43)

where

"" i\l' . I'L II/),," + g

/)"

o

l4

!~ ~~ (.) (.)

GaB = L.. LL O~NMoa(e) N M

HtU = LLL (~NM(rt

(e) N M

(2.45)

Equations (2.43) and (2.44) lead to a determinate system, whose fi nite-

element approximation is

~ - 3 -(Ka6 + kGae)A + fa = 0

a

where

K = 2:L (MIi) T H (N .r) TCia • a Ar Ci

Ii r

(2.46)

(2.47)

f = f + ~Ci Ci ~

b.

d MIi.r d M'an B' N ,G Q' an H• cx CXp

are defined in (2.45'.

::

III. MIXED VARIATIONAL .PRINCIPLES FOR LINEAR BOUNDARY-VALUI~ PROBLEMS OF

THE TYPE Au + ku + f = 0

In this section, we show that results ohtained previously using pro-

jection operators can also be generated usi.ng Ritz-type approximations of

certain functionals corresponding to various forms of boundary-value

..

l5

problems involving T*T eperators. We remark that the development of an

abstract theory of T*T operators was initiated by von Neumann [8J, and

Murray [9], and a general theory of approximate solutions of linear

equations and eigenvalue problems for such operators was given by Kato

[lOJ, and Fujita ell]. In recent times, the theory was extended and cast

into a general variational setting by Noble [l2J. See also the works of

Arthurs [13J, Robinson [l4J, and Oden [5J, and in particular, the recent

sUl1U11arypaper of Noble and Sewell [15J. Following ([l2J-[lSJ), we can

construct the bil inear functim al

..

(3. 1)

The funct ional .I (u ,y) in (J.1) involves both t he variables u and ~, and

can be used to define a "mix~d" variational principle corresponding to

(2.ll). The variational principle corresponding to primal problem (2.ll)

and dual problem (2.14) can be derived by assuming that(2.l2) and (2.13),

respectively, a re identically satisfied. Then we l\:lve the new functional s

l(u) = -21

[Tu,TuJ + [f,u} + ~(u,lI} + (B>'(Tu,g-u}~ - [13*s,lI}O\i1 2

('3. 2)

and

(3.3)

In elasticity, I(u) representH total potential energy, K(y) represents

total complementary energy, and .I(u,y) is the functional in the wcll-

known Hellinger-Reissner variational principle. Obviously, the functionals

I(u), K(y), and J(u,y) apply to more general classes of problems.

16

J(u,~ assumes a stationary value at the point (u·k,!.*) if and only

if u* is a solution of (2.12), and ~* is a solution of (2.l3), I(u)

assumes its minimum value at u*, the solution of (2.11), and K(~

assumes its maximum value at '!...>'(, the solution of (2.l4).

=IaThen

Let \1

These variational principles are related to the projections in an

a - ~ t:.a CPa' and ~_ = L.. b t:.!E. denote projections

t:.from (3.2), we obtain

interesting manner.

as described in section 2.

J(u,~ L L aaht:.[~ll,TCPQ'J

cx II

+ f L aCXaa(CPcx,cpa}+ L aa(f,CPcx} + L bll(B*!E.t:.,g}aR1

a,13 a II

- 2: I (B*!:£ll,cpcx}~l-L(B*S,CPCX}OR2

II cx cx(3.4)

OJ (aa b ) = 0-13 ' IIOll

2>ll([~ll,TCPeJ - (B*Qlll,cpe}~~l)+ k L aa(cpa'CPe}

II a

(J.S)

or

o (3.6)

which is, of course, identical to (2.24). In a similar fashion we can

obtain (2.18) from OI(aa)/oaS = 0, (2.2l) from OK(bll)/Oh, = 0, and (2.27)

afrom o.1(a ,bll)/obr = O.

IV. CONVERGENCE AND A POSTERIORI ERROR ESTIMATES

4.1 Convergence of Mixed Finite-Element Approximations. We now arrive

at a central topic of the paper: proof of the convergence of mixed

finite-clement approximation of boundary-value problems of the type in

17

(2.ll). Convergence and error estimates for pr~nal and dual prohlems

follow easily from associated minimum or maximum principles and are givcn

elsC\ ...here. Consequently, we confine our attention here to the mixed

problem.

To this end, let (U~'(,'i"() denote mixed finite-element SOltit Ion of

(2.11). By construction of the mixed model [2.23) and (2.26)], 11* and

V* satisfy the following orthogonality relations.

IT*y'* + kll* + f,U} '" 0 in R lB*(y'* - S) ,u}~ = 0, for V lJ ( l1\.G:>

o in b~ ; [B(g - U'Ir),~~ '" 0, for V y_ ( JCII1

(4.1)Let liS dcnote the errors in finite-clement solution hy

eu = u* - U*, v* - Y>'( (4.2)

We now show that each error tends to zero In some manner ;JS h approaches

Zero. The following telTnna lays the essential steps towards the cOllvergence

theorem.

LeJml1a4.1. 1.et .l(u,:'.) Iw the hilinear functional in ('J.l). Let (f,.1)

hold, and ilSStlllW that U'1r S;)l'jSfjl~S thl' houndary conditlun on Cili1. Then

tlw folluwing illcqllalitil's hold.

(i) .1(t1 , v*) ~ .1(\rk , v) , u E: '/.,{ [lIld v (' 'JJ'_ -.J

(i i) J (U, Y.*) ..~ .1 (U>'( , YJ ' II E ITlr. , V_ ( .)£1 (4. )

(iii) .I(u*,y'*) - .J(ll~'r,y) .I (1I'i( ''!...*) - .1 (D ,Y_'k)

~. To prove above inequalities we use the following relation, which

can be easily verified.

18

+ ~I\IUl - u211\<l -ill~l - ~1\2 + (B*(~ -S), Ul - 1I2}~

where 1\ I· III and 11'\1 denote the norms in'/..( and! respect ively, and

(4.4)

To prove (i), set III = u*, Y.1. = ~, \12 = u, and ~

(u*,~*) satisfy (2.11- 2.14), we obtain

v* in (4.4). Since

or

.1 (u.~*) ~ J (u*,y)

To prove (il), again we employ (4.4

(4.5)

= (T*V'k -I- klJ ~ f, U* - u 1+- [TlI* - V*, V - v* ]- . - --

+ (B*(Y.- V!), g - u* }C~il

By virtue of (4.l), we obtain

J(U*,Y) - .J(U,~*)

i. e.

(4.6'

..

The inequality (iii) is a trivial one in the light of above proofl:l. It

should he noted that in proving (i) we did not require u and v to satisfy

the boundary conditions.

Let (U,Y) denote interpolants of the exact solution (u*,'!..*), and h be

the maximum diameter of a finite element. Then the interpolant errors

lu* - u I and Iv>'<- V \ sati sfy the formulas as h ... 0- -

19

1°,/\1>'( - l'1) I ~ Chr, and \0i}'(Y.* - y> \ ~ nhH (4.7)

where C and D are constants independent of h, and r, B > O. lIere we hRve

used the ml1lti-integer notation: ~ = (0'1,0'2"" 'O'n)' 0'\ being integers;:; 0,

\ \ex 0' 0'

\ \' _ _ ~ 1 2. n _ ~

~ - 0'1 +0'2 + ... oj O'n' DO' - a lax1 ~2 , ... Oxn For example, U and V

are complete polynomials in ~_ of order p and q respectively, and T, T''< he

di fferential operators of order m S min(p +l, q + l),

Tu ~ ° u,~ 0'l~\~m ~

and T>''V = ~ (-1) \.@.\D v- a-IlSm -

(4.8)

Then we assume that U and V are stich that

C hP+11 '

(~ .9)

\.Je stat(~ the followl,ng theorem without proof (sec [6J), which gives

sufficient conditions for interpolant errors of the type (4.7) to hold.

Theorem 4.l Let u~..(~, ~. E &-l., be continuous, together with its partial

deri.vatives piecewise contil1\IOUS and bounded. Let U(~ helong to a family

3' r of approximations for which partial derivatives of order rare specl-

fied at least one node. Let '11(~ hl~ an r-conformahle Interpolant of \I'k(~)

which together with its partin} derivatives up to order r, coincides with

t:.1I>'''(~ ;Il each node x E r/.. Moreover. let J r contai n a suh family .if uf

functions whose \~\th partial derivatives arc p., I -I~\ - equivalent to

the corresponding derivatives of IJ cumpletl~ polynomial of degree p llnd let

r = p - 1; I~I :os; p. Then

(4. lO)

where C is a constant independent of h.

Theorem 4.2

20

Let the conditions of Lemma 4.1 hold. Then for all 0'1,0'2 )

o < 0'1 < I, 0 < 0'2 < k, and µ'1' µ':O>> 0

0'1I1ex..!!2+ ~ I lieu 1112 :;; 1I~*- y'1\2 +k \ lIu>'(- ull \2 -+ µ'lllT (u* - ii) 112 + ~ I \IT*(~* - yJI12

(4.11)

Proof. The proof follows d irect1y from Lemma 4.1, and the use of the

(i ii) leads to the result.

lIe.J\2 + k I lieu II \2 ::; \I~'( - yJI2 +k IlIu* - ull12 + 2\ITeu \I11~*- ill + 2I1eJ\IIT(u* - ti) \I

(f1.l2)

which with the help of above elementary inequality establishes (4.11).

Thus, theorem 4.2 establishes the fact that \\Ieu'dl"'O, \IeyJ\ ...O as

h"'O, provided interpolation errors of th~ type (4.7) hold. Specifically,

using (4.9) in (4.11), we obtain

Ci1Ilc!,.\12+Ci?\\Icul\\2 :'))lh2(Q+l) ±kClI12(\.t-V +C2µ'lh2(1'+1-1Il) +D2µ'2h2(Q+l-m)

(4.1))

It is clear that as h"'O the right hand side of (4.13) approaches zero,

concluding that \lleu~\\"'O and IloyJl"'o f.ndepcndently. We state another

important result in the following theorem.

Theorcm 4.3 Let (4.'3) hold, and assume that the interpolants (IT.,y,)satlsfy

the boundary conditions in (2.11). Then for all Ci) 0 < Ci < k, µ. > 0, we

have

•

I\C!,.1\2+ Q' \ IIeu 112S; \\~* - YJ\2 + µ. \ I\T*(u* - U) II \2

~. It is easy to prove that

J(U*,y'> S; J(u*,Y,*) S; J(U*,~*)

which implies

(4.14)

(4.l5)

(4.16)

21

Equation (4.l6) along with the elementary inequality leads to (4.14).

From the basic relations (4.1) we can establish a number of inequali-

ties that may be of usc in the future work. With this in mind we state

the following theorem:

Theorem 4.4 Let (4.1) hold. Then

(i) k IlIv)'( - vIII ~ I IIT*(~ - Y.*) 1\I + IIIT*y" kU + fll I

(ii) \l~- Y..*\\ ~ I\T(u* - U) \I + IITV - ~I\

(iii) \Ie -III. ~ II~''( - yJ\ + \\y* - YJI

(iv) Illeu III ~ Illtp'( - ull \ + II~* - £11

(4.17)

Partial Proof. Inequalitites (1) and (ti) follow directly from relations

(4.1), and (iii) and (iv) are trivial triangular inequalities that may

be employed in establish:l..ngconvergence for Y....*and U* independently. It

- -should be rcmarked here that we did not make any assumptions about (U ,YJ

and (U*,y....*),except that they Ratisfy (1.,9) and (4.1) respectively.

4.2 A Posteriori I':rrorEstimates. We conclude this section with lll(!

remark that the restilts of Aubin and Burchard [16J and Allhin (17J can

be extended to obtain a posteriori error estimates for the class of houndary

value problcms considered here. We state the following theorem without

giving the proof (see [l7]).

Theorem 4.5 Let

(i) [Tu , Tu ] ~ µ.\I \ u II I •

(ii) [Y..l'~] ~ Q'\bsll II~II(iii) U satisfy the boundary condition BU = Bg on OR1 and

y.. satisfy the boundary condition Il>'(~ = H*S on ~:;>.

22

Then the following posteriori estimates hold:

µ2 \ Ilu~( - ull\~ + k \ \lu* - u\l \2 S; k-1\ \If + U + T*y..\1\2 + 02µ-1 1Iy.- TUI\2(4. l8)

U + T*V\l \2 + <i'µ-1 \Iv - TU\l2- -(4.l9)

where u* is the solution of (2.ll) and v* is the solution of (2.l4).

v. AN I':XAMPLE

To demonstrate the ideas presented thllsfar, w~ work a simple but

representative example. Consider a rod of unit length, modulus E, .9nd

area of cross-section A resting on an elastic fOllOdation of modulus k, and

subjected to axial force f{x). The differential equation governing the

motion of the rod is

d2 \Idx2

.;- k \I + f (x) = 0, 0::; x S; 1 (S.la)

where u(x) is the axial displacement, k = k!EA, and rex) = F(x)/I':/I..

Suppose that the har is fixed at x = 0 Ilnd fre(' at the other end. Con-

sequently, till;! boundary conditiulHl nrc

u (0) 0, ~(l)dx

o (S.lh)

Without loss of generality, we take k

solution is

I, and f(x) ~ x. Then the exact

u(x) :s sinh(x)cosh(l) - x, (5.2)

To recast the problem (5.1) into the form (1.1), we define the inner

product 1

(H, v) =J Hvd.

o

(5.3)

23

and T = dldx so that T'l'( = -d/dx, and 13* ~ B =- 1 on ~ (x = 1) and

B* = n = -Ion ORl (x = 0), with &t = (O,l). Thus (S.l) is eq\livalent to

the canonical pair

v = Ttl, o on ~il

(5.4 )

T*v + kll + f(x) = 0, Il*v = 0 on 06t2

Note that in this example the functions g and S are prescribed to be

zero. \~e use the theory given in this paper to find an approximate solu-

tion to (5.l). For computational convenience we select same basis func-

tions for spaces U and!. The following set of linearly independent

functions (cp ) = (w~) are selected.er -

xcp (x) = 1 - - O:;x~h1 h

x- - (k - 2) (k-2)h~ x ~ (k-l)hh

cpk (x) = ~ (k = 2, J, ... , N - 1)x (k - l) h ~ x ~ khk - -h

xcp (x) = - - (N - 2) (N-2)h :c:x.,; (N-l)h (5.5)N h

where N is the total number of nodes; hence, the inte-rval [O,l] is divided

into (N - 1) equal parts, h = l/(N - 1).

a. Primal Problem. For O\lr example (2.18) becomes

o (5.6)

er

where Ncra, Gere

and feare computed using the definitions (2.l9):

24

1 -1 0 2 1 0

-1 2 -1 1 4 1

o -l 2 . 0 1 4

N 1\h I (5.7)

(~~X ~)h' Gaa = '6

(N X N)2 -l 0 4 1 0

-1 2 -1 1 4 1

o -1 11 I . 0 1 2

(5.8)

To satisfy the boundary condition u(o) = 0, we omit '1' which amounts to

deleting first row and column of above matrices. Figure 1 shows the exact

solutions against finite element solutions U* and v* for '(N-l)= N. = 4, where

v* is calculated from v* dU*dx

b. Dual Problem. Because of our choice of ~~, we have

and

and we compute ~tusing the definition (2.22),

1

2

2

·1' ~g - 2

2

2

1

('j.9)

-0.1

-0.2

-0.3

II,V

o 0.2 0.4 0.6

'r imn IN :0 I,

e

1.0x

/o:xacl sol'll ion

0- ~ -0 Fi nil e I':lement oll.l iOI.

-0.1

-0.2

-0.3

II,V

Dua 1Ne

4

Fiv,lIre 1. Comparison of primal and dllal fillite element solll ionswith the exact soilition

-0. J

-0.2

-0.3

U,V

-0.1

-0.2

-n.3

-0.1

-0.2

-0.3

Il,V

000

x

x

x

Finite Jo: I PIII"llt ~;oIlItit'll

Figllre 1. COIII),arison 0 I prililtl J and ·1.181 tillite e lClJlellt so ]I.tion!!"lit!: the exact sobltiOll

25

Again, to satisfy the boundary condition dU(l)=V(l)=O, we omit m, whichdx ~

amounts to deleting the last row and column of matrices in (5.7). The exact

('

solutions are plotted against the finite element solutions V*, Rnd U* = \ V*dx

for Ne = 4.

c. Mixed Problem.

to be

The matrices N'b. and M~ for ollr example> srI;;'complIr-edOt Ot

1 -l 0

1 O-l

o 1 0

12

o -l 0

1 0-1

o 1 1

(5. to)

The solutions obtained Ilsing mixed approximation are plotted in fig. 2

for Ne = 2,4,8. Logarithmic plot of the norms of the errors III e u I II and

\\e.!.l1 against the mesh size is shown in fig. 3. The rates of convergence for

u as well as for v = du/dx are the same. However, it is observ~d that

the error seems to oscilate beyond certain value of N, the nUIIILwrof sub-

divisions into which the domain is divided. This is caused due to the

numerical instability.

Remarks. Mixed finite-element approximation of a general class of boundary

value problems using the concept of projections is presented. Convergence

of mixed finite-element approximation is established, and a ntlmerical

example is presented. The numerical results obtained using mixed model

are in close agreement with the exact ones. Errors in hoth displacements

and stresses appear to be of the same order when the same types of inter-

polation functions are used in I;;'achapproximation. 'fInIs,the rate of

convergence for stresses is greater than that obtained by usil1R IlslIaldis-

placement type models. However, the error tend to oscillate beyond certain

•

o0.0

O.l

-0.3

u,vo

-0.1

-0.2

-0.3

0.2

0.2

0.4

0.4

0.6

0.6

0.8

0.8

N=2

N=4

x

x

r

•

u,vo

-0.3

u,v

0.2 0.4 0.6 0.8

000

x

Finite EJ eill/'Ill Su I.. t:j Ilil

Exact SIII',liu!;

tl,e exact sollltiulJ

•

7.5

7.0

6.5

p

~ 6.0

5.0

4.5

R ~/~\ / \

/ I h'Ef \/ I \ \

/1 \ 0\/ / /\ Q.I \ / \ ~I b' \ \

\ \

\ b\(:).

~\\

~

0.00.5 1.0

-In (h)

1.5 2.0 2.5 3.0

Fi/-,"ure 3. Rate or convergence for mixed fillite elemeut solljtiol~S

5 .•

•

2h

values of N, indic;lting the solution is unst:lble. Convergence of mixpd

finite-element approximation of abstract houndary value problems .'lnrl

s harper error estimates rl'mafn std)jects of futl.re invl!stigations.

Acknowledgement: Th~ support of this work by the U. S. Air Force Office

of Scientific Research under Contract F44620-69-C-0124 is gratefully

acknowledged.

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Equations, A. K. Aziz (ed.), Academic Press, New York (to appear).

..

•

27

6. .T. T. Oden, Finite Elements of Nonlinear Continua, McCraw-lillI, New

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l2. B. Noble, "Complementary Variational Principll's ror I\onndary Valuc

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13. A. M. Arthurs, Complementary Varialional Principles, Oxford University

Press, 1970.

l4. 1'. II. Robi.nson, "Complcmentary Variational Principles," Nonlinear

Functional Analysis <Illd AppllcaLions, I.. ILRall (ed.) Academic

Press, New Yor~, pp. ~07-57h, 1971.

IS. B. Nohle and M.. J.Sewl'll, "On lIual 1':xtrcllIlllll Principles in Applied

M<lthematics," .J. Illst. MathR. Applies, Vol. 9, pp. 123-193, 1972.

16 . .I. 1'. Auhin and II. G. Burchard, "Some Aspects of the Method o[ the

J1ypercircle Applied to Elliptic Variational Prohlems," Nllmeri.cal

Solution of Partial Differential Equations-II, SYNSPADE 1970, B.

Hubbard (ed.), Academic Press, New York, pp. l-67, 1971.

•

•

28

17. J. P. Aubin, Approximation of Elliptic Boundary-Value Problems,

Wi1ey-Interscience, New York, 1972 .

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CONVERGENCE OF MIXED FINITE ELEMENT APPROXmATIONS OF A CLASS OF LINEARBOUNDARY-VALUE PROBLEMS

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-Convergence and general properties of mixed finite element models of a

general class of boundary-value problems of the type

Au + ku + f c 0, U ( 6t

B (u - g) II:: 0 on ~l' B*(Tu - s) = 0 on oR2

are considered here where u = u(~ is a function defined on a bounded region R of En,OR is the smooth boundary of R, x is a point in R, A is 8 linear factorableoperator, k is a positive constant, and Band B* are operators describing mixedboundary conditions on oR.

UNCJ ASSTFIEDSecurity Cla •• lflcatlon

UNCLASSIFIEDSecurity cl... liic.t1on

14. L.INK .. L.IN K • L.IN\( CKEV WO"OI

"OL.. WT ROL." WT ROL.lL WT

CONVERGENCE

NUMERICAL ANALYSIS

FINITE ELEMENTS

BOUNDARY-VALUE PROBLEMS

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