s,

COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 11 (1977) 175-206© NORTH-HOLLAND PUDLISIHING COMPANY

MIXED-HYBRID FINITE ELEMENT APPROXIMATIONS OFSECOND-ORDER ELLIPTIC BOUNDARY-VALUE PROBLEMS

I. BABUSKA. J.T. aDEN and J.K. LEEInstitute for Fluid Dynamics and Applied Mathematics, University of Maryland.

and Texas Institute for Computational Mechanics. The UniL'ersity of Texas. Texas. USA

Received June I976

Mixed-hybrid finile element approximations are described for second-order elliptic boundary-value problems inwhich independent approximations are used for the solution and its gradient on the interior of an element and thetrace of the gradients on the boundary of the element. A priori error estimates are derived with conditions for con-vergence. Several other finite element models are also obtained as special cases. Some numerical examples are included.

I. Introduction

During the last decade three nonconventional finite element methods have emerged that havesimultaneously found wide application to a variety of technical problems and have also met withsome notorious failures owing to their delicate convergence and stability properties. We refer tothe so-called hybrid methods, the mixed methods. and the methods which use nonconformingelements.

1.1. Hybrid methods

]n 1964 Jones [1] proposed an interesting method of finite element approximations in whichreduced continuity requirements on the coordinate functions were made possible by treating ele-ment continuity conditions as constraints and carrying along Lagrange multipliers as new depen-dent variables. Similar methods were proposed also by Pian [2] and were subsequently expandedand extensively applied by him and Tong and ultimately dubbed "hybrid finite element methods"(see [2l-4], and references therein). Recently Raviart [5], Raviart and Thomas [6], and Thomas[7] outlined some interesting mathematical properties of a class of hybrid elements for secondorder problems that appear to fall into the category of the displacement models of Yamamoto[8] and the assumed stress mode] of Pian [2]. Brezzi [9] and Brezzi and Marini [ 10] studiedstress-assumed hybrid models for fourth-order problems. See also [] ] ] for some interesting obser-vations concerning variational formulations of hybrid models.

1.2. Mixed methods

The so-called mixed finite element methods were introduced by Herrmann [] 2], and their con-vergence properties were studied by aden and Reddy [13, 14], Ciarlet and Raviart [15], andJohnson [16]. In mixed methods, operators in a given boundary-value problem are decomposed

176 I. Babuska. J. T Oden and J.K. Lee. Mixed-hybrid finite elemellt approximations

into two or more parts. and the corresponding dependent variables are represented by independentfinite element approximations. For example, independcnt approximations might be used simu]-taneous]y for the solution Ii of Lap]ace's equation and for its gradicnt. and the result is often im-proved accuracy for approximations of grad II.

1.3. Nonconforming elemen ts

A characteristic of nonconforming finite element methods for boundary-value problems oforder 2m (in which the energy involves derivatives of order m) is that approximations to deriva-tives of order m-] may suffer simple discontinuities across interelement boundaries. Noncom-forming elements have also been investigated by several authors (see for example [] 7], 1181, [ 19]and [20]).

We mention that the engineering literature contains many variants of these methods. Somcquite general models have been suggested by Atluri [211 and Wolfe [22]. and extensive refcrcncesand computational accounts can be found in [3] or [23]. among others.

In the present papcr we develop a theory of mixed-hybrid approximations in connection with amodel second-order elliptic boundary-value problem in which both independent approximationsare used for the solution and its gradient on the interior of the element. and an independent ap-proximation is uscd for the normal derivative on the boundary. Thus. our method is simultaneous-ly a hybrid mcthod and a mixed method, and we show that either can be extracted from ourtheory as a special case. But what is also interesting is that it is a nonconforming method as wc]l.

2. Notations and preliminaries

Throughout this paper .n shall denote an open bounded Lipschitzian domain in a two-dimen-sional cuclidean plane with a piecewise smooth boundary an. The notation x = C\', . x 2} is usedto denote a point in n,a elifferentia] elcmcn t or n is denoted dx = d\' I dx 2' and an clemen t ofan is denoted d~.The boundary an is taken to be a collection of N smooth arcs, and the interiorangles (Xj of tangents to the arcs meeting at the jth joint are in the range 0 < (Xj < 21T, I ~ j ~ N.Polygonal domains are thus an important special subclass of the domains considered here.

For an integer m ~ 0 it is well-known that the Sobolev space Hm (n) is a llilbert space definedas the completion of the spacc of functions. infinitely diffcrentiable on n. in the Sobolcv norm

(2.])

•••

The inner product on 11''' (n) is denoted

I. Babllska. J. T. Oden and J.K. Lee. Mixed·hybrid finite elemellt appro;'Cimations 177

11. vE H"'(n) . (2.2)

We also usc interchangeably the notation 1I0(n) = L2(n). Likewisc.II~'(n) c HIt/(n) is the com-pletion in the norm 1I'lIm•n of the space of inl1nitely smooth functions with compact support inn. In addition, for 11/< 0 we define Hm (n) as the completion of L 2(n) in the norm

IfllvdxlIIl1l1m n = sup n .

• vEH-m(n) IIvLm.n11/<0. (2.3)

In much of the subsequent analysis we also deal with functions defined on the boundary an.Obviously. the notation L 2(a n) dcnotes the space of functions whose squares arc Lebesgue-integrable on an. It is also well-known that if II E /J I(n). the trace 'Yoll on an is well defined. Wedenote by H 1/2(an) the space of functions defined on a n which are traces of functions in HI (n).and we furnish 11112(an) with the norm

111f>1I1/2•an= inf {1I11lll.n:If>='YolI}.uEHI(n)

Clearly,

(2.4)

(2.5)

Conversely, there also exists a continuous map IJ of HI/2(an) into 1I J (n) such that

(2.6)

.....

Indeed. if v E Ii I(n) is sllch that

-~v + v = 0 on n ,

on an.

thenllvlI1,n = IIIf>IIJ/2.an·We recall that the Sobolev imbedding theorem establishes that [24]

(2.7)

(2.8)

where C(n) is a positive constant depending only on n. It follows that HI/2(an) C L2(an).Therefore, if 1/1 is a given function in L2(an), and if If> E HI/2(an). the integral Ion 1/I1f> cis hasmeaning. Consequently, we can introduce a norm 1I1/1L,/2.an on 1/1 defined by

178 J. Babllska. J.T. Oden and J.K. Lee. Mixed·hybrid finite elell/em approximations

I f 1/Il/> cis IllWIL 1/2 an = sup an

, tpEHI/2(an) 11¢1l1/2,an(2.9)

We now define the space H-I/2(an) as the completion of L2(an) in the norm II'" -1/2.an'The space H-1/2(an) plays an important role in later developments, and it is fitting to analyze :

its structure in more detail. First. we establish a lemma similar to one given by Babuska [25]:

LEMMA 2.1. Let 1/1E L 2( a n) be given, and let z E HI (n) be the weak solu tion of the Neumannproblem

-Az + z = 0 in n,(2.] 0)

aZ/an = 1/1 on an.

Then

1I1/111~1/2,an= f 1/I'YoZcis = IIzll~.nan

(2.1 ])

Proof This follows from the definition (2.9), together with (2.5) and (2.6). and the fact that zis uniquely characterized by

J [(V'z)V'v + zv] dx = f W'Yov cisn an

and

IIzlI~,n = f 1/I'Yoz cis. •

We easily see from these results that

1I1/1L1/2,an C;;; C(n) 1I1/1110,an '

(2.] 2)

(2.13)

(2.]4)

where C(n) is again a positive constant depending on n. Hence, L2(an) c f/-1/2(an).Some additional propcrties of 1I-1/2(an) should be noted. In particular, if ~(x) is a function

defined on an such that Hx) = I on some measurable subset fl c an and Hx) = 0 on an - fl'then 1/1~E L2(an) for every 1/1E L2(an) and 1I~1/I1I0,an EO;; 1I1/1110.an.On the other hand, if1/1E 1l-1I2(an), then, in general, ~1/1f/. H-I/2(an).

We shall also make use of another special space of functions defined on portions of the boun-daries of two-dimensional domains. Let a ni be a smooth arc of a n. Let v E Hm (n), m ;;;;.2. Thenavian E L2(ani). We now denote by Hm-3/2(an, n) thc space of all normal derivatives avian offunctions v E 11m (n) furnished with the norm

I. Babllska. J.T. Odell and J.K. Lee. Mixed·hybrid finite element approximations 179

i= I. 2 .... N}. (2.15)

.. We shall also deal with spaces of vector-valued functions. For instance. if lJ = {a t, a2} is suchthat aj E Hm (n). i = I, 2. we write lJ E Hm (n) and

(2.16)

Analogously, L2(n) = HO(n).Finally, we conclude our preliminary remarks by recording two theorems which playa funda-

mental role in later developments. Proofs of these theorems can be found in (261 (see also [27]and [28]):

THEOREM 2. J. Let CU and 'V be two real Hilbert spaces. and let B: CU X'V -.9i be a linear func-tional on CU X'V stich that for every II E CU and v E'V the following hold:

IB(II, v)1 ~ C(IIIIII'll.lIvl!ev ,

inf Stlp IB(II. v)1 ~ C2> 0 ,lIull'll= ( IIvllev" (

(2.17)

(2.18)

sup IB(u, v)1 > 0,uE'll

VE'V . (2.19)

Here C1 and C2 are positive constants independent of II and v, and 1I'1I'lt and 1I'liev denote thenorms on CU and 'V, respectively. In addition. let fE 'V' be given, where 'V' is the dual space of '17.Then there exists a unique element 110 E CU such that

B(IIO' v) = f(v)

Moreover.

VVE'V .

•

(2.20)

(2.21)

THEOREM 2.2. Let CUIr and 'Vir be finite-dimensional subspaces of real Hilbert spaces CU and ev.>l\ respectively, and let the bilinear fonn B: CU X ev ~9\ of Theorem 2.1 be such that for UE CUIr,

VE evlr• the following hold:

inf sup IB(U, V)I ~ C~ > 0 ,II U II'll = 1 IIV 11<).. <; (

(2.22)

sup IB(U, V)I > 0UE'll11

V:f: 0, VE'V Ir . (2.23)

180 I. Babllska. J. T. Oden and J.K. Lee, Mixed-hybrid finite element approximations

In addi lion, let f E CU ' be given as in thcorcm :2. I . Then(i) thcre exists a unique Uo E CUI/such that

B(Uo' V) = f( V)

with

IIUolI'll ~ (l/C~/)lIfll'll' ,

VVEC)JI/' (2.24 )

(2.25)

(ii) if Uo is the unique solution of (2.20),

lIuo - UoII'll ~ (I + C1!C:) inf lllio - iiI/'ll' •UE'll1l

(2.26)

Hereafter. C shall denote generic constants which are not necessarily the same in differentplaces; we shall always indicate the major dependence of such constants.

3. A model Dirichlet problem and a special variational principle

We shall concentratc on the following model Dirichlct problem:

-Au + II = f in n .(3. I)

on an.

where f E L2(n) and g E H1/2(an). As usual. a function u' E HI (n) is the weak solution to (3.1)if 'You' = g on an and

(u·. v)I,n = J fv dxn

(3.2)

We are interested here in a special formulation of a variational boundary-value problem- equivalent to (3.2) under certain conditions - which is designed to lead naturally to a theoryof hybrid finite element approximation. We begin by considcring a domain n == no C 912 of thetype described in section 2. Let P be a partition of n into a collection of E subdomains ne, 1,2, ...... , E(P) such that the following hold:

(i) E = E(P) < 00 ,

E(ii) n = u ne,

e=1

(iii) the subdomuins ne are Lipschitzian with piecewise smooth (COO) boundaries ane'

I. Babuska. J. T. OdclI alld J.K. Lee, Mixed·hybrid [illite clement approximatiolls

(iv) the boundary segments

181

e>[, OO:;;;e, fO:;;;E,

O~e.fO:;;;E.

are sets with a finite number µ(e.f) of components, and

where r:f are smooth (closed) an;s, p(e, f) 0:;;; µ(e, f), and Se f is a set of isolated poin ts such that

p(e.{)

Sef n U r:f= 0 ,g=1

In this way. we can define unambiguously the collection of smooth boundary pieces

r = r(P) =E

Ue·f=Oe>f

1~ g~p(e.{)

(3.3)

We assume that r is oriented so that ne is on the left side of r:r' e > f, E ~ e.[~ I.In what will follow we deal with integrals over ane which should be understood throughout in

the following sense:

N(e)

f v ds = 6 f VPi Idsl 'an ,=1 r.

e I

where ri, i = 1,2, ... N(e), are smooth pieces ofane which can be uniquely identified with r~f'Pi = +1 if ne is on the left side of ri, and Pi = -I if ne is on the right side of rio TIle orientationof r; is the same as r~f' e > f and the integral f(') Idsl is understood as a nonoriented one.

In addition, if v is any function defined on n, we shall denote its restriction to ne by ve. i.e.

We now introduce a space Hm(P) of functions defined on the sub domains ne' I 0:;;; eO:;;;E,endowed with the norm and inner product

[E JI/2

Illu IlIm,p = ~1 lIull~"ne

E

«u,v»m.P = 6 (lI,v)m.n ...e=1

Clearly,

11m (n) C Hm (P) , 111> 0,

182 I. Babllska. J. T. Oden and J.K. Lee, Mi.-..:ed·hybrid/inite element approximations

and if Ii E HO(P), thcn also Ii E Ho(n). We continue to use thc notation L2(P) = 11°(P).In addition. we introduce the space L 2(r) of square-integrable functions on r with the norm

I ~e~E, O~f~E, I ~g~p(e,f).

wherein 1/Ie denotes the restriction of 1/1E L2l!,) to one' Then, for any 1/1E L2(r). we define

E

1I1/111~(f) = .6 1I1/1"1I~1I2.an,, 'e=1

(3.4)

where CW(r) is the completion of the space L2(r) in the norm 1I'lIcw(r)'In (2.11), II1/ILI/2.an is defined with the aid of the wcak solution of the Neumann problem.

We now define ze E HI (ne) such that for a 1/1" = 1/Ilane' 1/1E L2(1'),

(Zeo V)l.ile = f 1/I,,'Yov cisane

(3.5)

Denoting by ne the unit outward normal to one. we have

Then, as in Lemma 2.1.

1I1/1ell~1/2.ane = IIzeII i.ne = f 1/Ie'Yoze cis.ane

Similarly_ we use the notation (see also (2. 15))

(3.6)

We remark that a given 1/1ECU?(r) can be restricted to one and that 1I1/1ell_ll2.ane ~ 1I1/1I1rW(r);however, the sct of restrictions of 1/1 E CU?(r) on one is not necessarily closed in H-1/2(ane).

Now, for a given partition P of n of the type described above we introduce the product space

together with the norm

E

= e~ lliliell~.ne + lI(J'ell~.ne + 1I1/1ell~l/2.ane] .

(3.7)

(3.8)

I. Bab/lska, J. T. Oden alld J.K. Lee. Mixed-hybrid finite element approximations 183

whcre A = {II, a, 1/1},II E HI (P), a = {o P 02} E L2(P), 1/1E CU7(r). It is also convenient to denotcby AI' the triple of restrictions {III'- aI" 1/Ie} to f2e and to lise the notation

IIAell~e = lI{ue,a 1" 1/Ie}lIie = lIuelli.ne + lIaell~.ne + 111/11'11:1/2.one .

so thatE

IIAII2 = 6 IIAellie .1'=1

Let BO., ).) denote a bilinear form on :x x:x given by

E

B(A, I) = 6 Be( AI" ie) .1'=1

where the bar denotes an arbitrary eh~lllcnt, and

l.s;;;e"E, (3.9)

(3.10)

(3. I I)

Be(Ae' ie) = floe,vue + Ueue + (Vue -aehi'e] dx + f (1/Ieue + ~elle) ds.ne one

The fundamental properties of this form are established in the following theorem.

(3.12)

THEOREM 3.1. The bilinear form B(· _.) of (3.1 I) satisfies all the conditions of theorem 2.1with the choice of constants C1 = 2 and C2 = 15-1/2 (hence, independent of the partition P) whenwe set CU= 'V =:x, where ex is the space defined in (3.7).

Proof: We first show that B(A. A) is continuous on :XX:X ; i.e. we show that (2.17) is satisfied.Recalling (3.5) and (3.6),

f 1/Ieue ds = (ze' Ue)l.ne" IIzelll.nelluellt.ne = II1/11' Lt/2.anellue IIt.ne 'one

Similarly,

l"e"E. (3.13)

j ~eue cis" lI~eLt/2,anelillelll.ne'one

(3.14)

Likewise, the use of Schwarz's inequality reveals that

f[ae'Vlte +Uelle + (Vile -ae)'ae] dxne

" IIaello.ne" VUello.ne + "uello.ne"ue"o.n I' + IIVllello.ne lIallo,ne + IIoello.ne lIaello.ne· (3. I 5)

Thus, by combining (3.13). (3.14) and (3.15) and successively using the Schwarz inequality, weobtain

184 I. Babllska. J. T. Odell and J.K. Lee. Mixed-hybrid finite elemellt approximatiolls

E

B{#., i) = 6 11e( Ae' i:e) ~ 211A:i'X II ill 'X ... =1

(3.16 )

Thus (2.17) holds for the bilinear form in (3.11). and Ct = 2 is independent of the partition P.Next, we verify that BO" i) satisfies the conditions (~.18) and (2, 19). Toward thi~ end, we

introduce for arbitrary l = {It. CJ, ljI} the special triple A= {II. a, .J;}. whcrc i_lne = A.e is given by

... .. ~ .Ae = {lie· CJe, ljIe} I ~e~E,

where ze is the solution of the local auxiliary problem for a ljI E CW(f'):

(~e' lie)(.He = f ljIeve d~ane

We easily verify that

(3.17)

Now we obtain directly from (3.12) the inequality

(3.18)

Be(Ae.ie)= f[(VlIe)'Vlte+211;+O'e'O'e]dt+ J[(VZe)'VlIe+Zelleldx- f ljIelleds+ f ljIezedsne n" ane ane

where we havc used (3. I 7). (3.5) and (3.6). Therefore.

H E

B(A, i) = 6 Be(Ae' ie) ~ ~ IIAelli,. = IIlll~ .c=l e=1

which. in view of (3. I 8), means that

BCA, i) ~ ~ II A. II 'X lIill<.x .viS

Thus.

inf supII A II'X =1 II A II'X <: 1

IB(l. kll ~ illfIIAII'X=I

.I BCA.A)I~ inf

II ill<:\: II A II'X = I

(3.] 9)

]n other words, condition (2.18) of theorem 2.1 holds. That condition (2.19) also holds follows

I. Babuska. J. T. Oden alld J.K. Lee. Mixed-hybrid /illite element approximatiolls 185

by interchanging the roles of A and A and noting that B(A, i') is symmetric. We emphasize that theconstants C, = 2 and C2 = 15-1/2 are independent of the partition P. This completes the proof. _

We next introduce a linear functional F on ex given by

E

F(i) = ?-J. [ J feue dx + f ge ~e Idsl] ,ne rK

eO

whereinfE L2(n) = L2(P), g E HI/2(an), ge = glreo'

(3.20)

THEOREM 3.2. The functional F: St'~ ffi given by (3.20) is continuous on the space St'definedin (3.7).

Proof: The fact that the first term is con tinuous on St' follows immediately from the Schwarzinequality. The second term requires some additional consideration.

Let ~ E CW(f) n L 2(r) be given. Then, in view of (3.5), there exists a z E HI (P) such that

(3.21)

Next, we take v = u·, where u· is the solution of (3.2) (recall that H I(n) CHI (P)). Thus,

E E

~ f ~ell; ds = f ~g ds = 1; f ge~eldsl,e-t an an e-I

e rKefand, according to (3_21 ),

f~gds=«Z,lt·))I.I'~ IIlzllll.1' 11111·111•. 1' ~ 1I~1I'W(l')lIu·lIl.n" 1Ii:1I~lIgll./2.anan

This completes the proof -

We are thus led to the following special variational boundary-value problem: find A. E St' suchthat

where B('#.., i) is given by (3.11) and (3.12), andF(i) is given by (3.20). By virtue of Theorem 2.1and Theorems 3.1 and 3.2 just proved, we immediately have the following:

t

- -B(A,).) = Fo.), v'#.. ESt' , (3.22)

THEOREM 3.3. There exists a unique solution '#..0 E St' of problem (3.22)_ Moreover, there existsa constant C> 0, independent of the partition P (indeed we can take C> 15-112

), such that

(3.23)

186 I. Babuska. J. T. Odell alld J.K. Lee, Mixed·hybrid finite elemel/t approximatiolls

The overriding question now is: what is the relationship between the solution). 0 of (3.22) andthe solution of our original model problem (3.1)? We resolve this question in the following theorem.

THEOREM 3.4. Let A.°E~ dcnote the solution of(3.22) and u· E H1(D.) the solution of theweak Dirichlet problem (3.2). Moreover, let }".E ~ denote the triple

}".= {u·, VU·. 1/1'} ,

where 1/1; = -au;/alle on r!f' e > f. Then

(3.24)

(3.25)

[Note that when 11· E H1(D.), au·/all has no sense in general. However, we define 1/1; here in theweak sense such that

f 1/I;'Yov ds = (u·, v)l.ne - 1fv dxone ne - - ~On the other hand, we know that 11· E H2(D.), where D. is arbitrary but such that D. c D.. Therefore,

au;/an has a normal sense on r - aD..]

Proof Introducing (3.24) into the definition (3. I I) and (3. 12), we get

E

B().·,k)= ~ [1[(Vu;)·VUe+u;ueldx+ f 1/I;ue ds+ f tI;~e cis]ne ane ane

E

= ~ [1feue d\:+ f Ue~e cis] .e-l ne ane

However. using thc definition of tane (. ) cis. we obtain

E

~ f tie ~e dx = J g~ cis ,e-I ane an

and, therefore,

(3.26)

(3.27)

B(},,' • i) = F(i)

Hence, (3.25) follows from the uniqueness of the solution ).0 to (3.22) •

Remark 3.1. Theorem 2.1 is still valid when CW(r) is restricted to any closed nonempty sub-space 9'(r) c CU9(r), and the constants Cj are not altered. Likewise, the functional F(l) of (3.20)is then also continuous, and }"o is again uniquely determined. Of course, 1/10 = 1/1. only when1/1. E T(r).

I. Babuska. I. T. Oden and I.K. Lee, Mixed-hybrid finite element approximations 187

..

Remark 3,2. The space ~of (3.7) can be restricted in a variety of ways, each of which will leadto a different type of variational problem. For example, we could use n I = no, E = I. and CJ = V ll .

Then we obtain the Lagrange multiplier theory of Babuska [25] (see also Babuska and Aziz [26]).When CJ::/= V 1I, we obtain a space used in the mixed finite element formulations of aden and Reddy[13, 14]. Still other special cases could be cited. We consider some of these in section 7.

4. l\1ixed-hybrid finite element methods

The stage is now set for the construction of finite element approximations of the variationalboundary-value problem (3.22). As expected, we will investigate a sct of partitions P of n intoE(P) subdomains which is now viewed as a finite element model of n; i.e. each domain ne.I ~ e ~ E, is now viewed as a finite element. The particular formulation that we have describedin the previous section provides a direct vehicle for the construction of mixed-hybrid finite ele-ments: ovcr each element we introduce local approximations of ue' CJe. and ljIe' using polynomialsof possibly differing degrec.

As usual, we associate with every partition P a parameter h such that

and

h = max he'l"e"E

I E;;; e E;;; E(P) , (4. I)

(4.2)

We also construct a collection of finite-dimensional subspaces over the partition which have thefollowing properties:

(4.3)

where 1'k,(ne) is a space of polynomials of degree E;;; k' over ne' '"For any function U E H'(ne) there is a constant C3 > 0, independent of he' and aU E Ql(P)

such that

where s = 0, I, I ~ I, and

17=min(k+ I -s, l-s).

Moreover,

I E;;; e E;;; E, 0 E;;; r E;;; r'} ,

(4.4)

(4.5)

(4.6)

where l'r' (ne) is a space of vector polynomials of degree E;;; r' over ne'

188 I. Babuska, J. T. Odell alld J.K. Lee. Mixed-hybrid !illite elemellt approximatiolls

For any (J E f/q(ne) therc is a constant C4 > O. indepcndent of he' and a i: E Q~(P) such that

wherein q ~ 0 and

w=min(r+ l,q).

Recalling that r!f are smooth arcs contained in r{P), we define

(4.7)

(4.8)

Q;-1/2(r) = {\V E 'U9(r) : 1/J~f = 'l11 g E ~r.(r~f)' t' ~ / ~ 0, E ~ e > f~ O}, (4.9)ref

where 'J>,,(r:/) is a space of polynomials of degree ~ /' on rff·For any 1/JE 'U9(r) () L2(1') so that 1/Je= 1/Jlane. and 1/J:f = 1/JlrK there exists a consklnt Cs > O.

independent of he. and a \fI E Q;-1/2(f) such that ef

where

(4.10)

() = minU + 3/2, 111 - I). m~ 2. (4.11)

In addition, we denote by Q;-1I2(ane) the subspace of Q;-1/2(r) consisting of functions whichvanish on r - ane'

Clearly. the product space

(4.12)

is a subspace of the space st' defined in <3.7).The inequalities (4.4), (4.7) and (4. I 0) are assumptions that we formulated. The question is:

under what conditions are these inequalities valid? It is possible to show, for example, that if allne are quasi-uniform triangles or quadrilaterals, then our assumptions are valid. We call an elementne quasi-uniform if ~lere exists a constant 00 > 0 indepcndent of P such that hJiie ~ 00,

I ~ e ~ E(P), where he is the diameter of the largest circle that can be inscribed in ne' We mentionthat there are many other cases where the assumptions are valid, e.g. curvilinear elements. We willassume that the partition is such that (4.4 )-(4. 12) hold everywhere.

The mixed-hybrid !inite element met/wei involves seeking an element

A= {U. 1:. 'l1} EQ

such that

B(A, A) = F(A) VAEQ (4. I 3)

where B(·, .) is the bilinear form deflned in (3.1 I) and (3. 12). and F(.) is the linear form in (3.20).

I. &l>uska. J. T. Oden and J.K. Lee. Mixed·hybrid finite element approximations 189

The existence of a unique solution to (4. I 3) depends upon whether or not the basic properties(2.17), (2.18) and (2. 19) of B(', '), established in theorem 3.1, are carried over to the approximateproblem (recall theorem 2.2). The remainder of this section is devoted to a study of these condi-tions for (4. I 3).

Let n~and n~denote orthogf)nal projections of ,!I(f'l.f,) ~nd~L2(f'l.e) onto Ql(f'l.e) and Q~(f'l.e)'respectively. and let us construct a special element A = {U. t. 'Ir} E Q sllch that

{

Ue = 2Ue + n~ze"_A/II. ~ A_ 0 01Ae - {Ue,te, 'lie} ~e - -I.e +ne( VUe)+ne("IIezl.J·

'Ir = -3'Ire e

where ze E Hl(ne) satisfies (3.5) for 1jJ = 'Ire E Q;-I/2(f'l.e)' We observe that

(Ze, Ue)J.ne = f q,eUe ds = (II~ze' Ue)J.neane

and, owing to the continuity of ng and II~,

(4.14)

(4.15)

l~e~E. (4.16)

Then, making use of the identities

..

we easily establish that

+ IIn~VUell~.ne - J (VUe)' (Vn~Ze - n~ Vn~Ze) dx.ne

To obtain proper bounds on these terms, we now introduce three basic parameters:

IInlz 112

= (QI (0 ) Q-I/~(af'l. )) = . fee I.neµe µe k e' t e ill 2

"'eEQ,II2(ilne> lI'Irell-1I2.ane

(4.17)

(4.18)

(4.19)

190 I. Babuska, J.T. Odell alld J.K. Lee. Mixed·hybrid /illite element approximations

'Ye = 'Ye(Qi(ne), Q~(ne» = SUp

VeE Qk(Ue)

II VVe -n~VVello.ue

II VVello.ne(4_20)

Then, using the elementary inequality ab EO; (a2 + b2 )/2 and noting that 0 EO; µe' ve' µe EO; I, we have

Be(Ae, Ae) ~ (ve - 'YeI2) IIUell~.ue + lII:ell~.ne + (µe - 'YeI2)II'1rell~1I2.aue ~ f3e(ne)1I'f'e112 e'(4.21)

wherein

(4.22)

Finally, summing (4.21) over all E elements, making arguments analogous to those we used in1l1eorem 3.1, and recalling Theorem 2.2. we arrive at the following approximation theorem:

THEOREM 4.1. Let B(A, A) denote the bilinear form on Q X Q in (4.13). Then

inf sup ImA, A)\ ~ I 5-1/2t3(P),IIAII =1 IIAIl =1

wherein

(4.23)

(4.24)

and f3e(ne) is detined in (4.22). Moreover, if f3(P) > 0 for a given partition P of n, then there existsa unique solution A0 to the mixed-hybrid finite element approximation problem (4. I 3), and

(4.25)

where A. 0 is the unique solution of (3.22), C 1 = 2 is the constant of continuity appearing in (3. I 6),and C~ = ] 5-112 t3(P) •

Remark 4.1. It is clear that in the special case that

Q~(P) ::> v(Qi(P» ,

which occurs whenever k' - 1 <;; ,.' (i.e. VUe E Q~(.ne) VUe E Ql(ne)), then

(4.26)

'Ye = 0 , (4.27)

In this case

(3(P) = min( 1, µ) and µ = min µe'l';;e<E

(4.28)

THEOREM 4.2. The parameter µe defined by (4. I 8) is positive if and only if the following condi-tion holds for any \}Ie E Q;t/2(ane):

,

I. IJabllska. 1. T. Odell and 1.K. Lee. Mixed·hybrid finite element approximations

and µ > 0 is a sufficient condition for the existence of a unique solution_We now give a necessary and sufficient condition for µ to be positive.

191

(4.29)

Proof: Suppose that µe = O. Then there exists 'lte E Q;"2(ane), 'lte =1= O. such that

Using (4. 15), we have

J'It U ds=OI.' I.'iH1e

which implies \}I I.' = 0 by (4.29), a contradiction.Now let µe =1= 0 and 'It I.' =1= O. Then. by (4. I 8) and (4.15),,

which is the contrapositive form of (4.29) •We refer to (4.29) as the rank condition because it obviously depends upon the rank of a

rectangular matrix produced by introducing members of the spaces QL(ne) and Q;l/2(ane) intothe contour integral. The same condition has been imposed by Raviart and Thomas [6 J. Discus-sions on this condition arc given in section 8 for a specific case.

Remark 4.2. If the inclusion (4.26) does not hold. Le. if k' - 1 > r', then it could happen thatve = O. For this case. assuming that the inverse property (cf. [27J p. 89)

1holds, we obtain from (4. I 7)

Be(Ae' AI.') ~ 2I1Uell~.ne + lII:ell~.ne + µell 'It I.'n:..1/2.ane - 'Yell VUello.ne 1I'i' ..."-112.ane

~ IIUell~.ne + (Ch2- 'YeI2) IIVUell~.ne + lI~ello,ne + (µe - 'YeI2)II\}1ell~l/2.ane .

The inequality (4.21) still holds for this case with the choice of (31.' as

192 I. Babllska, J. T. Odell alld J.K. Lee. Mixed-/lybrid [illite elemellt approximatiolls

Then the parameter ~(P) in (4.23) may still be positive so as to guarantee a uniquc solution of theapproximate problem. However, in view of the form of the estimate (4.25), the dependence of(3(P) on h may destroy convergence of the method.

Gearly, ~(P) > 0 is only sufficient for the existence of a unique solution to (4.13). However.we furnish a necessary condition in the following theorem:

THEOREM 4.3. In order that the approximate problem (4.13) have a unique solu tion, itis necessary that for a \{f E Q~)/2(f),

E

~ § \{f I' VI' cis = 01'=) anI'

implies that \{f = O.

VVEQk(P) (4.30)

Proof: Let there exist a unique solution to (4.13) for arbitrary data FECX' and let \{fo E Q~1/2(r)be such that \{f0 '* 0 and (4.30) holds. Then

B( {O,D. 'itO}. A) = 0 . VAEQ.

However, B(o, A) = 0, V A E Q, also. Hence, Q is not a unique solution for F = 0, a contradiction.

Remark 4.3, When the equation -Au = f in n is approximated instead of (3.1 a), one can showthat the inclusion (4.26) is also a necessary condition in addition to Theorem 4.3. For details onthis see [29].

5. The dependence of µ, v and 'Y on II

It is important that wc establish precisely how the paramctcrs µe' vI" amI 'Y" or(4.18), (4.19)and (4.20) depend upon the mesh parameter h. Toward this end, consider an open boundedLipschitzian domain r; c CJ2 2 of diameter II, 0 < Iz ~ I, and a fixed domain q defined by

9 = {x: x = x /h, x E (j }

so that

dia q = I

Using obvious notation analogous to (4.3), (4.6) and (4.9), we introduce the spaces

Ql(~) = {o: o(x) = U(x) E Qk«(j)} ,

Q~(q) = {i: f(x) = 1: (x) E Q~(~)} ,

Q~1/2(a~) = {~: ~(x) = \{f(x) E Q;1/2(a(j)} .

(5.1)

(5.2)

(5.3)

(5.4 )

(5.5)

I. Babllska. J. T. Odell alld J.K. Lee. flUxed-hybrid /illite elemellt approximatiolls 193

Here, for example. QkU)) = {U: U E 'l'k'( <J), I ~ k ~ k'} etc. We shall establish a collection oflemmas pertaining to properties of functions defined on the spaces described by (5.3). (5.4) and(5.5) and then apply these to the finite element subspaces described in the previous article.

Next, we introduce a Hilbert space H}:(~) endowed with the inner product

[ii, V]I,h =! [(vil)·vv+h2iiv] d~

wherein {i(x) = lI(x), x E 9. and V = Izv = h {a/ax I' a/a'\·2}' Likewise,

[]"'[) - [" ....1"21I I,h - ll. 1I I.h

and

(5.6)

(5.7)

(5.8)[]iIlli./r = f 1z2 [(VlI)'\"U + 1I2] dx/h2 = IIltll~.o;'r;

Now, let ~ E 0;-1/2(a &) be given, and let z, p and (f E H I (~) denote solutions to the followingauxiliary problems:

[z. W]t,h = f ~w cIS'O~

[/3, W]I,II = ~ q,w cIS -A(q,) J w d~'Or; r;

Iq, IV11,q = £ q,w cIS - A(q,) J w dx'O~ ~

wherein

(5.9)

(5.10)

(5.11)

f.... ....I', . I" g = (V ( • )) • V ( . ) d.\

~and (5.12)

We have thus selected A('l') so that (5.1 I) has a solution. i.e.

IA(~) d~ - i~cIS = 0 .a~

Likewise, in order that the solution q of (5.1 I) be unique. we also require that

We now come to the first of two lemmas:

LEMMA 5.1. Let z. q E HI(~) be the solutionsof(5.9) and (5.11). Then

.... 2 .... 2OZOI.1I ~ Iqll.q+~(/l).

(5.13 )

(5.14)

(5.15)

194

wherein

I. Babuska. J. T. Oden and J.K. Lee. Mixed·hybrid finite element approximations

(h) = (A2/h2) I d~ ,II

(5.16)

and A is defined in (5.12).

Proof: From (5.9) and (5.10) it is clear that z = p + A/h2• and, therefore, setting w = pin(5.10) gives

OPOr,h = [~p cIS- A f P dx = [E, p] I,h - A J Ii dol:a~ q ~

= 0 z 0 i./. - A J z d£ - A J Ii d.~ .II ~

By setting w = I in (5.9) and using (5. 12) and the fact that z = p + A/1l2. we also find that f~ z d\- =S1l(h)/A and f~ p dy = O. Thus,

0"'02 - 0" 2Z I.h - POI.h + 9'l (h) .

Now from (5.10) and (5.11) we see that

(5.17)

(5.18)

Thus, combining (5.18) and (5.17) gives (5. IS) •We denote by TIh and TI the orthogonal projections of H,:(q) onto Qfc(~J) with respect to the

inner products [ ., . ] I.h and I', ·11.g. respectively. Let

(5.19)

"'I '"Then, V W E Qk(Ci),

[t, WJI,h = f q,W cIS,a~

[~ , WJ I.h = £ ~W ell - A(q,) !W d~,at; g

IQ, WI.. = f... q,WcIS-A(~)1 Wdt,a~

with the normalization Ig Q d~ = O. Again, we easily verify that

z = ~ + A/1l2• 0202 = o~ 02 + S1l(h), I,h I,ll

(5.20)

(5.21)

(5.22)

(5.23)

I. Babllska. J. T. Oden and J.K. Lee. Mixed-hybrid finite elemelll approximations

and

A 2 .... 2IIZIIl,h ~ IQI •. + .9'l(h).

195

(5.24)

LEMMA 5.2. Let Z and Q be given by (5.19). Then there exists a constant C6 > O. independentof 11, such that

OZO:.h ~ (I - C6h2)IQlf.~ +9'(h).

Proof: In view of (5.21) and (5.22),

However,

(5.25)

where we haye used Poincare's inequality (IIQllo,q ~ C6IQII,~' C6 dependent on §but not on h.because f ~ Q cL~= 0). Thus,

from which (5.25) follows in view of (5.23) •We are fiQally ready to apply these results to a typical finite element of the type described in

the previous section:

THEOREM 5.1. Let ne, 1 ~ e ~ E, be a finite element and fie = {Xl': Xl' = xe/l1e. Xl' Ene.he = dia(ne)}· Let fie E HI(ne) be such that

I (Vqe)' V{Ve dS: = f ~eWe ciS - Ae(~) I\Ve cLxne ane ne

wherein

(5.26)

'" f ........I '"Ae = Ae(\fI e) = \fIe cL\: / dx ,

ane fie

(5.27)

..... ...

and \fie E Q;-1/2(ane). I ~ e ~ E. In addition, let TI denote the orthogonal projection of HI(ne)onto Q1(ne) with respect to the inner product I', '11 n . Then• e

(5.28)

(5.29)

196 I. Babllska. J. T. Odell alld J.K. Lee. Mixed·hybrid fillite element approximations

(5.30)

wherein C6 is a constant independent of he' while µe' ve and 'Ye arc the parameters defined in(4.18), (4. I 9) and (4.20), respectively.

Proof: Equations (5.29) and (5.30) follow immediately from the definitions (4. 19) and (4.20)and the definition of nc. Thus, we concentrate on (5.28). We have, by (5.20), (5.8) and Lemma 5.2,

f \{feZI' cis = IIZelli.ne = 0 Ze Di.he ~ (l - C6h~)ITl tilt,ne + st1.(Jle) ,Clne

On the other hand. from (2.1 I) and (5.15) ,

Thus

µe ~ inf'l1eEQ, II2(ane>

(l - C6h~)ITI qli.ue + !A(he)

Iqli.fle + !A(he)

Inequality (5.28) now follows from the elementary inequality blc ~ (b + a)/(c + a) for a, C ~ 0and b ~ c •

Theorem 5. I is important because we obtain bounds in the parameters µe' ve and 'Ye by study-ing the master element ne only. Using Theorem 5.1 and Theorem 4.2, it is easy to show that, forexample, when only quasi-uniform quadrilateral or triangular elements are used, then for all suffi-ciently small h, if the rank condition (4.29) is satisfied,

µ = min µe ~ {j > 0 ,Ic;ec;E

where {j is independent of h.A similar analysis can be made to study the dependence of µ on h by constructing a smooth

mapping of ne onto a master element q.

6. A priori error estima te

We now corne to the question of convergence and error estimates for the approximate schemeprescribed. The principal results follow easily from theorem 2.2 and the properties of subspaces(4.1 )-(4.12).

THEOR£M 6.1. Let 1I E HI(. ), I ~ 2, be the exact solution of (3.1). Assume that the parameter(3(P) of (4.24) is positive. And let A= {U, rt, \{f} EQ be the mixed-hybrid finite element solution.Then the following error estimate holds:

(6.1 )

J.BabuskiJ. 1.T. Oden and I.K. Lee. Mixed-hybrid finite elemellt approximations

where

197

e = {II - U, (VU - 1: l, 1/1- \{f} , Q = min{k.r + I. 1+ 3/2,1- I} ,(6.2)

in which 1/1= -all/3n on f!/, e > J: (3(P) is as in (4.24), and C3' C4, Cs are as in (4.4). (4.7). (4.10).

Proof, By combining the results of section 3 and theorem 4. I. we have, in accordance withtheorem 2.2.

lIeli E;; Ao Jnf 11)..- All EO; Ao {lUll - UIII\,p + II (J- l:1I0.p + 111/1- ~1I'l11(r)} ,AEQ

where U, 1:, and ~ are as in (4.4), (4.7) and (4.10), respectibely. Introducing (4.4). (4.7) and(4. I0) into (6.2) with the aid of (3.8).

(6.3)

from which we deduce (6. I) in view of (2. I 5) due to the fact that u E H/(P). 1~ 2 •In view of the estimate (6.1) and remarks made in section 4, if subspaces satisfy(i) the rank condition (4.29) and

(ii) the inclusion property (4.26),then the finite element solutions will converge at the rate of Q as in (6.2) if all r2e are quasi-uniform. However. if (i) is satisfied but not (ii), then in general A 0 ... 00 as h ... 0 - this will destroyconve.,£gence.

If n is convex and if (4.26) holds, then by applying the technique of Nitsche (see for example[27] or [28]), the following L2 estimate can also be obtained:

(6.4)

where C' is independent of h, and Q is as in (6.2).

7_Special cases

Several special cases can be deduced from our general theory by imposing various restrictions onthe spaces X and Q. A brif-f discussion is given below for each case.

7.1. Hybrid-displacement model} (primal hybrid model)

If we let CJ = Vu throughout the formulation, we obtain a hybrid model sinlilar to the one pro-posed by Yamamoto [8] and studied by Raviart and Thomas [6]. Naturally. we set

)98 /. Babuska. J. T. Oden and J.K. Lee. Mb:ed·hybrid /inite element approximations

~I = fll(P) X 'U9(r). (7. I)

The bilinear form is now

(7.2)

Following the analysis similar to that in section 3, it is easy to see that the bilinear form B I(', . )

satisfies all the hypotheses of theorem 2.1. Indeed, we can take CI = 2 and C2 = 6-1/2, where C.

and C2 are constants in (2.17) and (2.18), respectively. Therefore. there exists a unique },I ={u I, ljIl} E ~ 1 such that

(7.3)

where F(·) is as in (3.20).Denoting the corresponding finite dimensional space by

(7.4 )

(with the same notations as in section 4), we have the following theorem:

THEOREM 7.1. There exists a unique finite element solution Al E Qt to (7.3) if the rank con-dition (4.29) is satisfied.

Proof: Following the same analysis as in section 4, we can take

(7.5)

where C~'is as in (2.26), and µe is as in (4.18). By Theorem 4.2 we see that C~'> 0 if (4.29) holds.This proves the theorem •

The necessary condition is the same as in theorem 4.3. Finally, we obtain the following errorestimate in view of theorem 7.1 and theorem 2.2:

THEOREM 7.2. Let the rank condition (4.29) hold. Then the following error estimates holds:

(7.6)

where

C¥( = max{k, t + 3/2, I-I}. I'? 2,

in which C~, C3, and Cs are as in (7.5), (4.4) and (4.10), respectively.We comment here that the estima te (7.6) slightly di ffers from one in [5] which reads

(7.7)

I. Babllska. J. T. Oden and J.K. Lee, Mixed-hybrid finite elemem approximations

where T = minCk, t + I), the obvious reason being the difference in norms on r.

7.2. j\'lixed method with a constrained boundary condition

199

This is a generalization of the Lagrange multiplier method studied by Babuska [25]. We beginby setting

S(2 = H1(D.) X Lin) x H-1/2(an),

Q2 = QL(n) x Q~(n) x Q~1/2(an) ,

where

Q1cn) = {U E HI (n): Ue E l'k.(ne), I E;; e E;; E} ,

Q~(n) = {1: E L2(n): 1:e E '}J r.(ne), I E;; e ~ E} ,

(7.8)

(7.9)

(7. I0)

(7.11 )

(7. I 2)

Note that Q1(n) is CO-finite element space that does not satisfy the boundary condition. Thecorresponding bilinear form on S( is

B20., i) = f [a.VU + uu + (Vu - a)·a] d~ + f (1/!u + 1/!u) cis.n an

(7.13)

As clearly indicated by the choice of spaces, the partition prescribed in section 3 is no longerneeded to study the variational problem associated ·with the bilinear form (7.13). Therefore, thedependency of variables and parameters on ne will be dropped throughout the analysis. Thissimply means that all the analysis made in sections 3, 4 and 5 is valid when we set E = I andnl = no = n throughout.

Finally, assuming that the rank condition is satisfied, we obtain the similar error estimate

(7.14)

where

e = {u - U, a - 1: , -au/anlan - 'l'} ,

a2 = min {k, r + I, t + 3/2, I - l} ,

in which

I ~ 2 .

l3(n) = max(O, min(v - -r/2, µ - -r/2)) .

200 I. Babuska. J. T. Oden and J.K. Lee. Mixed·hybrid !inite element approximations

Here the paramcters µ. 'Y and v arc dcfined in analogy to those in (4.18)-(4.20) with the index edropped.

A further obvious restriction would be to take a = V II throughout. Then the method is exactlythe one studied by Babuska [25].

7.3. Mixed method

If we assume that the boundary condition is satisfied a priori and if we set

(7.15)

(7.16)

whcrc Qk(n) is the usual CO-finitc elcmcnt subspacc which satisfies the boundary condition lindQ~(n) is as in (7.11). then the corresponding bilinear form on S(3 will be

B( OJ. w) = f I(J • ~ Ii + IIIi + ( V II - (J). G] dx .n

A slight modification of our analysis leads us to the following estimate:

where

e = {II - U, V II - 1:} ,

0:3 =min{k.r+ 1./-1}.

(7.17)

(7.18)

in which v is as in (4. I 9) with e dropped. For details of this method see [9] and (10).Obviously. by letting IT= V1I. we obtain the usual conforming displacement method, and the

estimate is

(7. I 9)

where

0:4 = min {k. / - I}. /~ I .

8. The rank condition

We discuss here the rank condition mentioned in section 4 in connection with a specific example.

/. IJoburka. J. T. Oden and J,K. Lee. Mixed-hybrid finite element approximations 201

In particular, let the domain n be quasiuniformly triangulated. and let r denote a master elementwith the boundary ar. Further. let k = k' and t = 1': that is Q~(r) = 'J>k(r) and Q;1/2(ar) = 'J>r(ar)are spaces of complete polynomials of order k over rand t over each side of a r. respectively. Then,since the rank condition is invariant under coordinate transformation, the condition (4.29) can berewritten for a \{I E 'J>r(ar) as

!\jIUds=OilT

implies \{I=O. (8.1 )

The following theorem, proved by Raviart and Thomas in [6]. also applies to this particular aspectof our problem:

THEORtlH 8.1. The rank condition (8.1) is satisfied ifand only if

It + I

t+2

if t is even,

if I is odd •For furthcr discussions on criteria such as this, consult [6].

Remark: The sufficiency condition holds for the case when k' > k but not thc necessary condi-tion.

Finally, we mention that the rank condition for the case discussed in section 7,2 takes a bitdifferent form: namely. for a given \}I E Q;t/2(ar2),

implies U=o. (8.2)

And it is casy to show that this condition is satisficd if and only if k ~ t. provided that no morethan two sides of ar coincide with a n. Here r need not be a triangle.

9. Numerical experiments

As a simple examplc designed to verify our theoretical estimates. we consider the following one-dimensional problem:

I-UII + 11 = [(x) .

u(O) = 11(1 ) = 0 .

where

xEn=(o.l).(9. \)

[(x) = 20:(1 + 0:2(1 - x)(x - x)) + (l _x){tan-1 a(x - x) + tan-1 ax} .

- (1 + 0:2(X - X)2)

202 I. Babllska, J. T Odell alld J.K. Lee. Mixed-hybrid fillite element approximations

1.60

1.20

0.80

0.40

0.000.40

Fig. I. Exact solutions.

1.20

in which a and x are some given constants. The solution [30] is

It(x) = (I - x) (tan-1 a(x - x) + tan-J ax) . (9.2)

The solution has a sharp knee near x when a is large (see Fig. I). We consider the following twocases:

I. Smooth solution: Q = 5, x = 0.2,2. Rough solution: a = 50, x = 0.4,

both solutions are shown in Fig. I.Following the notation in section 3, r now consists of E + I knots; hence, eWer) = ffi E+ 1 and

1/1 E CW(r) is an (E + I)-tuple

where 1/1; are real numbers definea at each knot Xi' 0 ~ i E;;; E = 1/11. According to Lemma 2.1,(3.4) and (3.6), we have

(9.3)

and one can show that

(9.4)

I. Babl/ska, J. T. Oden and J.K. Lee. Mixed-hybrid finite elemelll approximarions

where c does not depend on h.The corresponding spaces are

st= H1(P) X L2(P) X CW(r) ,

and

203

Clearly, the rank condition (4.29) is satisfied if and only if k ~ I. Hence, if k ~ I, the estimates(6.1) and (6.4) can be written as

{lIlel/lIl.,p, lIeollo.p. lIe.µIICU/(r)} ~ C·/2Q(·)

and

lIellllo.p '" CC·hQ+1

(.),

(9.5)

(9.6)

-16

-I

I-al-I-

'"-8

-4~Iope : 2

o Mixed-hybrid I'lethod (k=2.r.: 1)

o Displacement method (CO-quadratic)

o Divergent solution (k·2.r=O) in h

-2

k=2. roO,

2

-3 -4

Fig. 2. Convergence curve for toe smooth problem.

204 I. Babuska. J. T Odell alld J, K. Lee. ilfixed-hybrid fillite element approximatiolls

where 0: = min{k. r+ I}, C is a constant indepcndcnt of h. and C' is such that\. CO is independcnt of II if r ~ k - I (see rcmark 4.1).2. C· = C't 1+ C"h-2) ifr < k - I (see rcmark 4.2),Numerical results show that the actual rates of convergence are as the estimates derived with the

exception of l<'y1max. wherc the adual rate is better than what follows from (9.4). This is in com-plete agreemcnt with thc behavior of the usunl finite element method where thc rate in l'lmax isbetter than the theoretical estimate obtained by simple application of thc imbedding theorem.This effect h<lSbeen completely proven by Nitschc I 17] very recently. So we can cxpect that theabove-mentioned effect for the behavior of le",llIIax could be theoretically explained. Figs. 2 and 3show that the <lpproximation is convergcnt for the case k = 2. r ~ 1 and divergent when k = 2.r = 0 for smooth and rough problems, respectively. Also shown for comparisons of accuracy aresolutions obtained using the mixed-hybrid method and the conventional conforming finite mcthodusing CO-quadratic elements. As can be seen from Figs. 2 and 3. accuracies in approximating Ii andIi' arc about the same for both models in an average sense. Ilowevcr, as anticipated, the approxi-mation '11 of Ii' at the knots is quitc accurate compared to that or the C°-quadratic element. Thiscould be an advantage of the mixed-hybrid method at the expense of a bit more manipulation.

For both thc mixed-hybri.d and displacement models. the interval is divided into E equal cle-ments. where E = 5. 10. 15.20.25.30. To insure sufficiently accurate numerical integrations. a5-point Gauss quadrature rule is used. and values of 1<,lmax are computed for thc displacement

-10

- 5

...o......'"

o llixed-hybrid method (k'2,r:: 1)

<> 01sp IaCel'1ent "'et~od (Co -quadrat ic)

o Divergent solution (k;2,r;l))

-4 in h

5

Fig. 3. Convergence curve for the rough problem.

I. Hab/lska. J. T. Odell alld J.K. Lee, Mixed·hybrid /illite elemell( appro.y:imatiolls 205

model at 2 Gauss points. The very slow convergence of le;,lmax in fig. 3 is evidently due to steepgradient of the solution and might have been improved if nodal points were placed tactically.

No meaningful comparisons of computing cost could be made because the computer programfor the mixed-hybrid method is specialized in this particular problem while the other is a quiteversatile one.

Acknowlcdgemcn t

The support of this work by the U.S. Air Force Office of Scientific Research through Grant74-2660 and the Atomic Energy Commission through Gran t AT( 40-1) 3443 is gratefully acknow-ledged.

References

[II E. 10nes. A gcncralization of the direct-stiffness method of structural analysis, AI AA J. 2 (1964) 821-826.[2] T.H.H. Pian. Element stiffncss matrices for boundary compatibility and for prescribed boundary stresscs. Proceedings of the

First Conference on ~Iatrix Methods in Structural Mcchanics. Wright-Patterson Air Force Basc. 1965. AFDL-TR-66-80 (1966)457-477

(3 IT.H.II. Pian and P. Tong. Ilasis of fmite element methods for solid continua. Int. 1. Numer. Meths. Eng. I (!969) 3-85.[41 P. Tong. New displacemcnt hybrid finite elcment models for solid eontillllOl. Int. 1. Numer. Mcths. Eng. 2 (] 970) 73-85.(5) P.A. Raviart.llybrid finite element mcthods for solving 2nd ordcr elliptic cquations. in: 1.1. Miller (cd.). Topics in Numerical

Analysis II. Proc. of the Royal Irish Com. on Num. Analysis. 1974 (Acadcmic Press, N.Y .. 1976).[6] P.A. Raviart and 1.M. Thomas. Primal hybrid finite element mcthods for 2nd order elliptic equations. Report 75025. Univ.

Paris VI, Laboraloire Analyse Numerique (1975).17] 1 .M, Thomas. Methodcs dcs clements finis hybrides duaux pour les problcmcs elliptiques du second-ordre. Report 75006.

Univ. Paris VI et Centrc National de 1a Recherche Scientifique (1975).[81 Y. Yamamoto. A formulation of matrix displaccment mcthod. Technical Report. ~lass. Inst. Tech .. Dcpt. ACTOn. Astron.

(Nov. 1966).(9) F. Ilrc7.zi. Sur la methode des c1emcnts finis hybrides pour Ic problemc biharmonique. Numer. Math. 24 (975) 103-\31.

(101 F. Ilrczzi and L.D. Marini. On the numeric'll solution of plate bcnding problems by hybrid methods. RAI RO Report (] 975).[Ill B. Fraeijs dc Vcubcke. Variational prindples and the patch leSI. Int. 1. Numer. ~Icths. Eng. 8 (1974) 783-801.(12 J L. Herrmann. finite elemcnt bending analysis for platcs. 1. Eng, \Iech. Div. ASCE 93 (1967) 000-000.(13) 1.T. Oden and J .N. Reddy, On mixcd finitc clemen! approximations, SIAM 1. Numcr. Anal. 13 (1976) 392-404.[14] 1.N. Reddy and 1.T. Odcn. Mathematical thcory of mixed rinitl' clement approximations. Q. App\. Math. 33 (1975) 255-280.[15] P.G. Ciarlet and P.A. Raviart. A mixed finitc clement method for thc bih'lTmonic equation, in: C. de Boor (ed.), Mathematical

aspects of finite c1cmcnts in partial differential equations (AL'3dcmic Press. N .Y .. 1974) 125-145.[16] C. lohnson. On the convergence of a mixed finite-element method for platc bendin~ problcms. Numer. M;llh. 21 (1973)

43-62.[171 1. Nitsche. Convergence of nonconforming methods. in: C. de Boor (ed.), ~Iathematical aspe~ts of finite elements in partial

diffcrential equations (Academic Press, N.Y .. 1974) 15-53.(18) I. Babuska and M. Zlama!. Nonconforming elemcnts in the finite clement method. Tech. Notc BN-729. lJniv. Maryland

(1972).(19) T. ~liyoshi. Convergenl"C of finite clement solutions rcprescnted by a non-conforming basis. Kumamoto 1. Sci. Math. 9

(1972) 11-20.120] G. Strang. Variational crimes in the finite elcment mcthod. in: A.K. Aziz (cd.). ~Iathematical foundations of the finite

element method with applications to partial differential equations (Academic Prcss. N. Y .• 1972) 689-710.[21] S. Atluri. A ncwassumed strcss hybrid finite element model for solid continua. AIAA 1. 9 (\971) 1647-1649.[22J ),1'. Wolf. Gencralized hybrid strcss finite clemcnt models. AIAA 1, I' (1973) 386-388.(23) 1.1'. Wolf. Gencralized strcss models for finitc c1cment analysis. Ph.D. Thcsis, Swiss Fed. Insl. Tech. (ETlI). Zurich (1974).1241 S. Agmon. Lectures on clliptic boundary value problems (Van Nostrand, 1965).125) I. Babllska. The finite c1cment mcthod wnh Lagrange multipliers. Numer. ~tath. 20 (1973) 172-192.

206 I. Babllska. J. T. Oden and J.K. Lee. Mixed·hybrid finite element approximations

[2611. Babuska. Error bounds for finite element method. Nurner. Meth. 16 (1971) 322-333.[27 J I. Babuska and A.K. Aziz. Survey lectures on the mathematical theory of finite elements. in: A.K. Aziz (ed.). The mathe-

matical foundations of the finite element method with applications to partial differential equations (Academic Press. N.Y.,1972) 3-359.

[28 I 1.T. Oden and J.N. Reddy, An introduction to the mathematical theory of finite elements (Wiley Interscience. New York,1976).

[291 J.K. Lee, Convergence of mixed-hybrid finite element methods, Ph.D. Dissertation. Div. Eng. Mech., Univ. Texas at Austin(1976).

[30 J H.H. Rachford and M.r:. Wheeler. An N"'l-Galerkin procedure for the two-point boundary-value problem, in: C. de Boor (cd.).Math. aspects of finite elements in partial differential equations (Academic Press. N.Y., 1974) 353-382.