....

A Mathematical Theory of HybridFinite Element Methods: Accuracy,Rates of Convergence, and Criteriafor Their Successful Use

by J. T. ODEN

Division of Engineering Mechanics, Texas Institute for Computationalt\-lecJtanics,The University of Texas. Austin, Texas

ABSTRACT: The purpose of this paper is 10 summarize and to elaborate on several recentresults obtained by Babuska, Oden and Lee (1-5) on the accuracy and convergence ofcertain types of hybrid finite element methods, and to show how these remits can be used to

design acceptable hybrid elements for second- and fourth-order boundary-value problems.

I. Introduction

Hybrid finite element methods have gained in popularity due to the work ofPian and Tong [e.g. (6)] and they appear to have some advantages overconventional finite elements in certain types of problems (e.g. plate- andshell-problems. problems involving stress singularities). The basic idea is torelax the usual interelement continuity requirements on the dependent vari-ables by treating these as constraints. The Lagrange multipliers introduced toenforce these constraints are generally identifiable with various normal deriva-tives of the dependent variable at element boundaries. In this way, theseboundary values (the Lagrange Multipliers) can be approximated indepen-dently and the proper continuity. in this reduced sense. is automaticallyprovided.

The history of hybrid methods is not unblemished; in some applications,apparently reasonable choices of shape functions for hybrid elements have ledto ill-conditioned or singular stiffness matrices. As a result of this unpredictablebehavior, we rarely find hybrid elements used in large-scale commercial finiteelement programs. We show that these deficiencies in hybrid elements can beattributed to their failure to satisfy what we shall call the rank condition.Fortunately. this condition can be tested a priori, so that it should be alwayspossible to avoid the type of difficulty described in future hybrid applications.

In terms of practical guidelines for constructing good hybrid methods, ourresults can be summarized as follows:

(1) The first point is rather obvious. Suppose that for the model problem

-tlu+u=f in nII = 0 on an (1.L)

413

1. T. Oden

(1.2)

where A = V2 and H is a smooth domain in IR2 with a smooth boundary an, wetreat the condition of continuity of the "displacement" u across interelementboundaries as a constraint. Then the Lagrange multiplier l/J is associated withilu/i)ll~ (i.e., the tractions) on the boundary ilOe of each finite element. e =1, 2 ..... E.

(2) With the choice of unknowns indicated in 1, there is a natural "energyspace" UUto which the solution pairs (u, t/J) belong, and the square of the normof a pair of functions in this space (equivalently, the energy of a pair. theinterior displacements and the boundary tractions) is given by

lI(u, l/J)I~= J1 {In. (IVuI2+u2)dX+IIl/Jllt-,n(oo.)}

-where the last term is a special boundary norm defined in (2.13).

(3) If hybrid approximations of (1.1) are constructed such that the approxi-mate displacements contain complete polynomials of degree ~k and theapproximate tractions contain polynomials of degree ~t, and if the approxi-mate problem is solvable for smooth data f, then the optimal order of accuracyachievable by the method in the energy norm UUof (1.2) is

lI(error in u. error in l/J)llou= O(hO') (1.3)

(1.5)

(1.4)

H'-2(H) [see (2.5)],

where II is the maximum diameter of a finite element in the mesh and

(}'= min (k. I +~).(4) However, if the data is not smooth. but is only in

then the error is, at most, 0(h'-1).(5) The numerical stability of the hybrid method [indeed, the very existence

of a solution to the equations approximating (1.1)], depends upon the behaviorof a special paramcter µ. defined as follows:

Let '1'. be an arbitrary approximation of the boundary traction l/Jon clementfl. and let z. be the exact displacement produced in an unloaded element dueto '1'.

Let Z. be an approximation of Ze over He using the same N shape functionsc!>i(x. y) used for approximating u. which is given by

Z. =f (r (Vze' Vcf>i + zA)i) dx dy)c!>i(X, y).,-1 Jo.

Then the stability parameterµ.. is defined by

= . f IIZell:,I(o.) (1 6)µ.. 10 II 112 .'1'0:0,1,'2 'I'e H-1J2(OO,)

where O;J is the set of boundary polynomials of degree ~ I and the indicatednorms are defined in Sect. II [see (2.4) and (2.13)].

(6) In order that the hybrid approximation problem have a unique solution,it is necessary that, for every finite element and all II,

µ'e> 0 e = 1. 2..... E ( 1.7)

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A Mathematical Theory of Hybrid Finite Element Methods

Moreover. (1.7) holds if and only if the following rank cO/ldition holds:

J 'IF,.U,ds=O V'u~EOhne) implies 'lFc=O. (l.8)"fl,

Physically, this condition asserts that the virlllal work done by a given approxi-mate boundary traction 011 allY approximate boundary displacemelll is zero if amionly if tlte tractions are zero. We discllss this condition in more detail in SectionVII.

(7) Similar results can be obtained for more general boundary-value prob-lems. such as fourth-order problems or the equations of plane or three-dimensional elasticity, and for other types of hybrid methvds. In each case.different energy spaces must be identified and the a priori error estimatesassume different forms, but one always encounters the question of numericalstability in the form of stability parameters sllch as µ". in (1.6) and the behaviorof these parameters can be assessed using rank conditions such as (1.8). Formore complicated hybrid models. these conditions assume more complicatedforms.

We comment on results analogous to (1.3) and (1.7) for other types of hybridmethods in subsequent sections.

II. Some Special Sobolev Spaces

The theory of linear elliptic partial differential equations is intrinsicallyrelated to the idea of Sobolev spaces, and such spaces form the fundamentalfiber of existence and regularity theory and approximation theory of ellipticboundary-value problems. An excellent account of properties of variousSobolev spaces can be found in the recent book of Adams (7); a summaryaccount is given in the recent book of Oden and Reddy (8).

Let 0 be any open bounded domain in two-dimensional euclidean space IR2.

We recall that L2(n) denotes the Hilbert space consisting of (equivalenceclasses of) Lebesgue measurable functions whose squares are Lebesgue integra-ble over n. The norm of a function II E L2(O) is, of course,

lIullL,(o)={L lul2dXfwhere dx = dXI dX2.

Now the notion of Sobolev spaces provides a systematic way of measuringthe regularity of the types of functions we usually encounter in variationalproblems and boundary-value problems. For example. a function II is said tobelong to the Sobolev space H'"(n) if its (generalized) partial derivatives(u, o 14/i)x " i)U/(JX2 •... , a"'u/i)x'2') of degree ~/ll are in L2(n). We write thisconcisely as

H'"(!l) = {u: D"1/ E L2(!l),lal ~ Ill}

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J. T. Odell

where multi-index notation is used:

(2.3)Ct., Ct2 = integers ~ O.

The space H"'(n) is a Hilbert space with inner-product and norm given by

(II, V)Hm(O) = r 2: D"'uD"v dx; IluIIHm«(}) = [(u, It)Hm(O)y' (2.4)In 1"'1"''''Since two functions in L2(n) are equal (almost everywhere) if they coincideeverywhere but on a set of measure zero. the derivatives appearing in (2.2) and(2.4) are to be interpreted in a generalized sense. Indeed, if U E H"'(n),classical derivatives of u of order 111 may actually not exist at a countablenumber of points in n.

In addition to the Sobolev space H'"(O), we introduce the subspace H;;'(O)of functions which, together with their normal derivatives of order ~ m - 1vanish on the boundary an of n:

H~'(n) = {u E H"'(n): D~u(x) = 0,x = (XI, X2)E i)O, j ~ m -I}. (2.5)

The topological duals of the spaces HO'(n) are of great importance in thetheory of elliptic equations. Recall that the topological dual OU'of a Hilbertspace ou. is the space of continuous linear functionals defined on ou.. The dualsof the spaces (2.5) are the so-called negative Sobolev spaces, denoted

(2.6)

Since the elements of H-"'(n) are linear functionals on Ho"(n), it is clear that asuitable norm on H-"'(O) is given by

(2.7)

The reason for the terminology in (2.6) is made clear by a simple example.Consider the step function u shown in Fig. l(a). Obviously II is square-integrable. but derivatives of u are infinite at the jump points. Hence liE L2(n)but Dlt == du/dxe L2(n) (here n = IR). Thus. we may denote L2(n) = HO(n).Likewise. the function v in Fig. l(b) may be differentiated once to produce anL2-function. whereas IV in Fig. 1(c) may be differentiated twice to produce sucha function (e.g. take u = Dv = D2

1V). Hence, v E H1(O) and IV E H2(n). Now qin Fig. l(d) is a collection of Dirac deltas represented by arrows: indeed, takeDu = q. Thus, q must be integrated once to give an L2-function. In otherwords. D-'q E L2(n). Hence q E H-1(n). We remark that this example [whichis taken from Oden and Reddy (8)] actually makes sense because of theSobolev embedding theorem, by virtue of which H"'(n) c CO(n). the space ofcontinuous functions on n, whenever m> 11/2, n being the dimension of n.

416 Journal of The Franklin Institute

A Mathematical Theory of Hybrid Finite Element Methods

w

IeHl(n)

(e)

~v

~eH'(,{l)

(b)

V

(0) 1111111111111

"""""",. eHo(n)

FIG. 1. Example of elements of the spaces Hm(D). -1 ~ 111 ~ 2.

Thus, for n = 1, the functions in H I(n) are continuous and 'l( v) = v(O) is notonly meaningful. but implies that q E (H1(!l))'.

From what has been said, it is clear that

(2.8)

(2.9)

In fact, this inclusion is dense and continuous.Boundary spaces. In studying boundary-value problems, we cannot escape

the necessity of describing the regularity of functions defined on the boundaryan of a domain n. It so happens that such studies lead naturally to the notionof fractional Sobolev spaces.

The most important relationship between the spaces Hm(n) and Sobolevspaces of functions defined on an is provided by the trace theorem [Lions andMagenes (9)]: Let n be a sm.ooth domain in IRn and 'YiLt denote normalderivatives at aO of a function U E Hm(n):

i)iu I'Ylu=j , O~j'5 m - Lan on

n being the outward normal on an. Then the 'Yi can be extended to continuousoperators mapping Hm(!l) onto Hm-I-t(i)D.); i.e .. there exist positive constantsq, independent of u, such that

lI'Yiullu"-J-,n(oll) ~ Cj IluIIHm(O), 0 ~ j ~ m - 1. (2.10)

What are the spaces H"'(a!l). C1i = m - j -!? One way of defining these

Vol. 302. Nos. 5 & 6. November/December 1976 417

J. T. Oden

boundary spaces is to introduce the norm on functions cP defined on an:(2.11 )

Then H"·-H(iln) is defined as the completion of the space of boundaryfunctions L2(i)n) with respect to this norm.

The reason that Sobolev spaces of fractional order occur naturally in dealingwith traces of functions defined in Sobolev spaces of integral order is atechnical one. If, for example, U EH1(n) and an is smooth, the largest numberµ. such that the extension "YoU of u onto i)n is a continuous map from H1(n)onto HI' (i)n) is µ. = t this can be shown using Fourier transforms [for acomplete account. see (8)]. The trace Yo is certainly a bounded map fromHI(n) onto HO(an)and there exists an entire "scale" of Hilbert spaces HS(i)n).o ~ s ~ 1. It so happens that the smallest of these for which yo is continuous isprecisely H~(an).

Again, we associate negative Sobolev spaces with the duals of the boundaryspaces:

H-rn+i+J(a!l) = (H"'-d(i)n))'.

For example. H-~(an)= (H!(i)n))' and

(2.12)

(2.13 )

Ill. Sobolev Spaces Defined on a Partition

With the notations and concepts described up to now in mind, we next turnto a special class of Sobolev spaces which has particular relevance in finiteelement theory. Let the domain 0 c IR2 given a partition P such that it isdivided into a finite number E of subdomains Oe such that

F.

fi= U fi,,: il.nn,=0. e¥f.,,=1

(3. I)

For simplicity. we shall assume that n and the subdomains ne are polygonal,although the extension of subsequent results to cases in which the O. arecurvilinear is not difficult. For example. P may correspond to a triangulation ofo such as that shown in Fig. 2.

Next, we denote by r the total interior boundary of n consisting of all of thesubdomain boundaries an. but excluding the corners (vertices) of the polygons!le. For example, if each n. is a simple triangle, we have

3

an. = U r~+{xN}~=I' 1:=5 e ~ Ei=l

4 18 Journal 0' The: ':ratnklin Institute

A Mathematical Theory of Hybrid Finite Element Methods

x'

r~Xl A"

r:'F,G. 2. Mesh geometry.

where r~denotes the open segments connecting the vertices xN. Then

r= U n.lS;~sEIsis3

We define the following Hilbert spaces on the partition P:(i)

H"'(P) = {u : II, = IIln, E H"'(n,). 1~ e s E}

IIIIIIH-(p)= {JllllI,II~I'"(O')}!(ii)

(3.2)

(3.3)

(3.4)

(3.5)

W(f) = {l/J : t/1, = l/Jlan,E }1~(i)n,), 1 ~ e s E}

{E 1 }!

Ill/Jllw(n = e~1 II"'ellw'12(oo,)

We shall see in subsequent discussion that the spaces H"'(P) and W(f) playacritical role in hybrid finite element methods.

IV. Some Hybrid Variational Principles

The spaces H"'(P) and W(f) described in the previolls article can be used toconstruct so-called hybrid or mixed-hybrid variational principles. The basicidea is to construct a local "principle" for each element n~and to then sumthese over the entire partition P.

We consider the following model problems

I. Membrane on an elastic fOlllldation

-Au+all=p(x,y)/T in O(a>O, T=consL).

II = 0 on an

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J. T. Oden

II. Torsion of a prismatic bar

II = 0,

( a2 il)-AII=206=f in n A=-+- .ilx2 ay2

II = 0 on anIII. Plate on an elastic foundation

A2u + l.W = p(x, y)/D in ni)2u~ = 0 on i)n (simple support).an

(4.2)

(4.3)

Obviously, more general problems could be considered. but these are adequatefor getting across the major concepts

Problem 1. Let us first consider problem 1. We begin by introducing specialbilinear and linear forms defined on pairs of functions (u, .p):

wherein

E

B«u, .p), (it, .J;)) = L Be[(ue, .pe), (ue, .J;.)]("=1

E

F(i~, ~) = L Fe(iie, .J;e).=1

(4.4)

(4.5)

(4.6)

FAiie, .J;e)= i fiie dx dy (f= pIT).o.

(4.7)

It is easily seen that B(· .. ) and F(') are defined on the product space

rJfl = Hl(p) X W(f) (4.8)

where Hl(p) and W(f) are defined in (3.2) (with m = 1) and (3.5), respectively.Indeed,

B :OUxUU,-+ IR, F:rJfl-+ IR. (4.9)

For simplicity in notation, we shall denote the pair (u, .p) (consisting of theSaint- Venant stress function and the shear stress .p = au/ane tangent to "in-terelemenf' boundaries ane) by A.:

Then the functionalA. = (II, t/J).

J(A.) = tB(A., A.) - F(A.)

(4.10)

(4.11)

represents the total potential energy of the membrane, and it assumes astationary value whenever A. is chosen so that A.. = A.lo. = (uln" i)u/al1.). and u is

420 Journal or The: Franklin Insilluk

A Mathematical Theory of Hybrid Finite Element Methods

the exact solution of (4.1). Thus, (4.1) is equivalent to the variationalboundary-vallie problem: Find (u . .p) e GILsuch that

8[(11, .p). (ii. cii)] = F(ii. cii) 't/(ii . .j;)E au.Problem II. The torsion problem leads to the hybrid energy space

'Y = L~(P) x W(r)

(4.12)

(4.13)

where Lj(P) is the quotient space consisting of equivalence classes of functionsin Hl(p) differing by constants on P:

E

111I11~i(P)= L IIVII.IIL(o.).e-1

(4.14)

(4.15)

We again arrive at a variational problem of the type (4.12) on 'V of (4.13) witha B(· .. ) given by (4.6) with a = 0: Find (1I, .p) e 'Y such that

B[(lI, .p), (ii, cii)] = F(ii, cii) 't/(ii. cii) e 'Y. (4.16)

(4.17)

Problem III. For fourth-order problems such as (4.3) it is natural to considermixed-hybrid variational principles: i.e .. we decompose (4.3) into the system

A IV = m in 0, W = 0 on an}Am+aw=f in n, m=O on an

so that instead of the energy space OUof (4.8) we are now led to the space ofquadruples

(4.18)

The variational principle is a four-field principle, involving interior displace-ments and moments, wand m, and boundary slopes 8. and shears .p.: Find~=(w, m, 8. .p)e~ such that

(4.19)where

8(<I>,~) = t {r (aw.w. + m.nl. - Vw•. Vm. - Vmp • Vwe) dx dye=-l Jo.,

+1 (8e111"+ii.m.+.p.lv~+.j;.W.)dS} (4.20)00.

F(~) = t r f.w. dx dy. (4.21).-1 Jo.Hybrid principles of this type were first studied mathematically by Lee (4).

We shall cite some of Lee's results in a later section.

Vol. 302. Nos. 5 & 6. No\cmbcrlDccember 1976 421

J. T. Oden

V. Existence and Uniqueness

While practitioners of mechanics generally have little use for existencetheory of partial differential equations (on the grounds that solutions automati-cally exist for any mathematical model constructed from sound physical argu-ments), existence theory, nevertheless, forms the cornerstone of approximationtheory. The approach to an approximation theory that accentuates existencetheory is this: determine the conditions which must be met if a given problem is tohave a solution; then determine whether or not these conditions are passed on tothe approximate problem. The degree to which existence conditions are fulfilledin the approximate problem depends upon the intrinsic properties of theapproximate method itself, and is intimately connected to the idea of numericalstability.

It is not by accident that all of the hybrid variational principles described inthe previous section fall into a very similar and general form suggested by(4.12), (4.16), and (4.19). This, in fact, is the general form of linear variationalboundary-value problems on Hilbert .spaces, and, despite this abstract setting,it is possible to make some precise statements about the solutions of suchproblems, their existence, uniqueness, and continuous dependence on the data.

Conditions for the existence of unique solutions to the linear boundary valueproblems of mathematical physics are given in the following theorem:

Theorem 5.1. Let UUand 'Y be real Hilbert spaces and B: au. x 'V - IR abilinear form such that

B(u,v)~Mllull'lLllvll'Y VueUU and Vve'Y

inf sup IB(u, v)I~'Y>Olullo.-1 Iv ...."'1

sup\B(u,v)I>O v#OueOU

(5.1 )

(5.2)

(5.3)

where M and l' are positive constants. Then there exists a unique elementuoeUU such that

Furthermore,B(uo, v) = F(v) Vv E 'Y. (5.4)

(5.5)

This theorem is a generalization of the classical Lax-Milgram theoremproposed by Babuska (10). A detailed proof is also given in (11) and in (8).Condition (5.1) is merely a requirement that B(', .) be continuous; (5.2) and(5.3) are called weak coercivity properties and insure that the operator as-sociated with the abstract variational problem (5.4) has a continuous inversedefined on its range. Then the solution llo will depend continuously on the dataF. as indicated in (5.5).

While the proofs are not trivial. it can be shown that the variationalproblems (4.12), (4.16), and (4.19) involve forms satisfying the conditions ofTheorem 5.1, that. therefore, they have unique solutions, and that these

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A Mat/lematical Theory of Hybrid Finite Element Methods

solutions are also solutions of the corresponding problems (4.1). (4.2). and(4.3). Proofs of these assertions are given in (1) and (4).

VI. Hybrid Finite Element Methods

We pass immediately to hybrid finite-element methods by constructingGalerkin approximations of. for example, (4.12) on subspaces of au. consistingof functions A = (U, 'IT) whose restrictions to the subdomains n" now viewed asfinite elements, are polynomials. Specifically. we introduce the spaces

O~(P)={U:Ue=Ulo.EPdne); l~e~E} (6.1)

O;-!(r) ={'I': 'I'd = 'lT1r-E P,([f): i~e~E, 1~i~3}. (6.2)

In other words, we consider pairs A = (U, 'IT) of functions whose restrictions toa given element fle are polynomials of degree ~k on the interior of ne and ofdegree ~ t on the boundary segments [f.

It is well known that such spaces are endowed with special interpolationproperties [see. e.g. (8)]:

(1) If U E Hr(n), r > 1. there is a (; E Qt(P) and a constant C1 >0. indepen-dent of P and u, such that

Illl- (;IIH'(0.)~C1h~IIUIIH'(0.)} (6.3)µ=min(k,r-l): h. = diameter (n,)

(2) Let .p belong to the space H"'-~(n.,i)Oe) which is obtained as theclosure of the space of normal derivatives av/alle e L2(iln,.) in the norm11.pllfi~-m(o..oo.)= inf {IIvllw'<o.) (m =:::2), avlane = .p}. Then there exists a 'irEO;-!(f) and a constant C2 >0, independent of P and .p. such that

11.p- 'l'llw,n(oo.) ~ C2~0' \I.pIlH--m<n ••oo.)} (6.4)(}'= min (1+2, m -1)

The hybrid finite element method applied to problem (4.12) consists ofseeking the element A,

such thatB(A. A) = F(A) VAeUU"

(6.5)

(6.6)

where B(A. A) is the bilinear form defined in (4.4) and F(A) is given by (4.7).Similarly. the space UU" is also a subspace of au. and can be used in Galerkinapproximations of (4.16).

For the variational problem (4.19). we introduce the space

~" = OL(p) x Q~(P) x Q;-!(r) x Q;-!(r) c ~ (6.7)

which leads to the mixed-hybrid Galerkin method for fourth-order problems:find 4> E~h such that

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VII. Numerical Stability and the Rank Condition

The question of paramount importance at this point in the analysis iswhether or not the approximate problems (6.6) or (6.8) have unique solutionsfor a given P and, if so. are these solutions bounded in the proper energynorms as It ~ O? We shall outline the answer to this question for problem I(i.e. (6.7)), and cite analogous results for problems II and III (i.e. (6.8)).

It is clear from Theorem 5.1 that (6.7) has a unique solution in au" wheneverthere exists a constant I'h >0 such that

inf sup IB(A,X)I~I'h>O. (7.1)IIAI""=1IlAI"'s;1

Condition (5.1) automatically holds since OUh C OU,and (5.3) holds whenever(5.2) holds because UU= 'V and B(', .) is symmetric.

To test condition (7.1), we proceed as follows:(1) Pick an arbitrary A = (U. '1') E °U" and formulate the special auxiliary

problem (we take a = I for simplicity)

-Aze+ze=O in ne

i)ze = 'I' on ane.i)ne

(7.2)

(7.4)

(7.5)

Physically, Ze represents the displacement in an unloaded membrane elementdue to the approximate boundary traction 'Jr. The total virtual work performedin this element due to stresses produced by Z moving through an arbitraryapproximate interior displacement Ue is

l 'I'eUeds=(ze, Ue)H1(1l.) = (7T1Ze, Ue)H'(O.) (7.3)Io.

where 7T1 is the HI-orthogonal projection of Ze onto o~(ne)' Clearly, the

virtual work done by 7T 1Ze is

117T Izell~'(Il.) = f 'I'e7T IZe ds.Do.

(2) Pick a special element A EOUh such that• 1Ae=(2Ue+7T ze.-3'J1,)

Then(7.6)

where C1 is a positive constant independent of he' and, from (4.6).. i 1 2 IBe(A~.A ..)= (VUe·V(2Ue+7T Ze)+2Ue+Ue7T Ze)dxdyo.

+l {(2Ue + 7TI z,)'Jre -3 Ue'l',} dsto.

= 2 II Uellti(O.)+ II7Tl Zellt1(0,). (7.7)

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A Mathematical Theory of Hybrid Finite Element Methods

(3) We would like to have the norm IIAI~on the right side of (7.7). Towardthis end, we introduce the stability parameter

_ . f lin 1z.lIt,(C1.)

µ'e - III II 112 .'V.EO,'nColl.) '1'. H-,n(Ml.)

Then, because O~µ..~1 and (7.6).• 2 2

B.(A •. A.) ~ 211U.IIIII(o,)+ µ'e 11'1'eIlW'12(oo,)

~ µ..IIA.II~ ~ ~: IIA.II'U IIAII'U·Thus.

(7.9)

inf sup IB(.\., Ae}1~ cµ"·UA.11..=1 1A.a..Sl 1

(4) In view of (7.1), we conclude that the mixed-hybrid method (4.12) has aunique solution so long as

µ'e>O Ve=1.2 .... ,E. (7.11)

(7.12)

(5) As a convenient test for determining if given subspaces lead to stabilityparameters µ'e satisfying (7.11), we note that in (1) the following condition isproved: the !'arameter µ.. of (7.8) is positive if and only if, for any 'l'e E Q;i(r)

J 'l'eU,ds=O VUeEQl(p) implies '1'.=0.00.

Condition (7.12) is called the rank condition, since it depends upon the rank ofa submatrix obtained by introducing members of Q;-i(r) and Q~(P) into thecontour integral. The physical interpretation of (7.12) has already been notedin item 6 of the Introduction. Moreover, when (7.12) is satisfied, µ.. isuniformly bounded below by a positive constant µ.* as 11~ 0 [see (1)]. Similarcomments apply to the torsion problem (4.2) (or 4.16)).

(6) Turning now to the mixed-hybrid method (4.19), we find that a necessaryand sufficient condition for the existence of a solution to (4.19) is, once again,that (7.12) holds. If different orders of polynomials are used for approxima-tions of wand nt, then some additional stability parameters must be intro-duced. For more details on such mixed-hybrid models, see (1).

VIH. Error Estimates

We conclude our study by establishing a priori error estimates for the hybridmodel of problem I and the mixed-hybrid method for the four-order problemII. These estimates provide a basis for selecting shape functions for such finiteclement methods.

A priori estimates can be obtained by a direct application of the fundamentalapproximation theorem for Galerkin methods [see, e.g. (10). (11). or (8)]: If Udenotes the Galerkin approximation of the solution of the abstract variational

Vol. 302. Nos. 5 & 6. No,·emberlD.""mber 425

(8.1)

(8.3)

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boundary-value problem (5.4) in some closed subspace UU"of ou.. then the errore = II - U satisfies

Ilell<\L S (1 + M) Ilu - Ullou)'h

where )'h is the approximation of )' of (5.2) in au" and U is an arbitraryelement of OIL".

Thus, hy direct application of (8.]) and use of the interpolation properties(6.3) and (6.4), we arrive at the following bounds:

Problem I. If the solution u* of (4.12) belongs to H~ln)then

IIA * - A *1I'll ~ CIl' 1I11IlH'(il) (8.2)

where A..~ = (ll~. all~/ane)' A* is the hybrid approximation of A..*, C is a positiveconstant, and

v = min (k. t +1, 1- 1).

Problem II. If the solution w* of (4.19) is in H/(D.), then

IIcfJ*- tP*lI~ C211"[JIwllt/(p)+ IIAwlli.,'-2(p)]iwhere .p*-tf>* = (w*- W*, m*- M*, 8*-0*, .p*-'l'*), C2 is positive constantand

l3=min(k. t+t 1-3). (8.5)

We remark that error estimates for Problem II (4.) 6) involve bounds of thesame order as those in (8.1). Also, Lee (4) has shown that L2-estimates can beobtained for Problem Ill, and these are very useful in comparing variousmethods and in choosing appropriate polynomial bases.

Acknowledgement

The support of ,this work through the U.S. Air Force Office of ScientificResearch under Grant 74-2660 is gratefully acknowledged.

References

(1) l. Babuska. J. T. Oden, and J. K. Lee, "Mixed-Hybrid Finite Element Approxima-tions of Second-Order Elliptic Boundary-Value Problems", (to appear).

(2) J. T. aden and J. K. Lee, 'Theory of mixed and hybrid finite element approxima-tions in linear elasticity". Lecture notes in mathematics, Proc. IUTAM/IUMSymp. 011 Applicatiolls of FllllClional Allalysis to Problems of Mechallics. Marseil-les, Sept. 1975. Springer. Berlin. 1976.

(3) J. T. aden. "Some new results on the theory of hybrid finite element methods".Lecture notes in mathematics, Marltematical Aspecls of rile Fillite ElementMeltlOds. Rome. Dec. 1975, Springer. Berlin. 1976.

(4) J. K. Lec, "Mixed-Hybrid Finite-Element Methods for Fourth-Order Problems",(to appear).

426 JournaJ of The franklin Institute

A Mathematical Theory of Hybrid Finite Element Methods

(5) J. T. Oden and J. K. Lee, "Mixcd-hybrid methods for the analysis of potentialflow". Finite Element Methods in Flow Problems, ICCAD, MicroLito. Genova,1975.

(6) T. H. H. Pian and p, Tong, "Basis of tinitc clement methods for solid continua",1m. J. Numerical Methods in Engllg" Vol. 1. pp. 3-35. 1969.

(7) R. Adams. "Sobolev Spaces", Academic Press, New York, 1975.(8) J. T. Oden and J. N. Reddy. An Introduction to the Mathematical Theory of Finite

Elements. Wiley-Interscience, New York. 1976.(9) J. L. Lions and G. Magenes, Nonhomogeneous Boundary-Value Problems.

Springer, Berlin, 1972.(10) I. Babuska, "Error bounds for finite element methods", Nllmeric/te Mat/tematik,

Vol. 16. pp. 322-333, 1971.(11) I. Babuska and A. K. Aziz, "Survey lectures on the mathematical foundations of

the finite element method". The Mathematical Foundations of the Finite Ele-ment Method with Applications to Partial Differential Equations, Edited by A.K. Aziz, Academic Press, New York, pp. 5-359. 1972.

Vol. 302. No•. 5 & 6. No'emberlDec:ember 1976 427