A THEORY OF MIXED FINITE ELE~lliNTAPPROXIMATIONS

OF NON-SELF-ADJOINT BOUNDARY VALUE PROBLEMS

J. T. Oden

Seventh U. S. National Congress of Applied Mechanics

Boulder, Colorado

June, 1974

"A Theory of Mixed Finite Element Approximations

of Non-Self-Adjoint Boundary-Value Problems"

J. T. Oden

1. Introduction

Some features of this paper may seem to be a bit too

technical for a general exposition on some aspect of mechanics,

as th~s should be, but the subject of the paper is at the

heart of much of the work on computer applications in solid

mechanics now-a-days -- namely, mixed finite element approxi-

mations. It is well known that by using.independent approxi-

mations for stresses and displacements in linear elasticity

problems, for example, improved accuracy in the stresses can

sometimes be obtained. For this reason, mixed models have

been used in a variety of recent applications in plasticity,

viscoelasticity, and fracture mechanics.

The real utility of such mixed models, however, has al-

ways been a cloudy issue. It is not difficult to find examples

of mixed finite-element approximations which lead to unstable

or ill-conditioned equations or which, despite a great deal

of additional labor, produce results which are inferior to;

or no more accurate than, those obtained ustng conventional

"displacement" methods. The need for an analysis of some of

- I -

the intrinsic mathematical properties of mixed approximations

has, consequently, been recognized for some time. A first

step toward fulfilling that need for a special class of self-

adjoint problems has been provided in some recent papers

[1,2]. There a theory is developed for the decomposition of

an operator of the form T*T into a single pair of canonical

equations which are approximated using Galerkin methods. In

the present paper, that theory is expanded to include non-

self-adjoint problems and to the more delicate question of

multiple decompositions in which any number of dependent

variables is approximated simultaneously. In some instances,

the results are quite surprising.

2. Decomposible and Non-Self-Adjoint Operators

Most linear problems in solid mechanics involve self-

adjoint operators of some type, and are, therefore, especi-

ally receptive to approximation by variational methods.

Finite element or Galerkin approximations of self-adjoint

elliptic problems, for example, lead to symmetric and usu-

ally well-conditioned matrices. Frequently, the variational

statements of such problems involve coercive bilinear forms,

guaranteeing the. existence of unique solutions which depend

continuously on the data, and, therefore, containing an in-

herent degree of stability. In recent years, the mathematical

theory of finite-element approximations of linear, self-

adjoint, elliptic boundary-value problems has been developed

j

quite extensively, and convergence criteria, error estimates,

criteria for selection of basis functions, and criteria for

numerical stability are fairly well known.

The corresponding theory of approximation of non-self-

adjoint problems is another matter, particularly when the

operators involved are not elliptic. The variational formu-

lation of such problems does not involve the conveniently

stabilizing symmetric, coercive bilinear forms, and the cor-

responding Galerkin approximations lead to unsymmetrical

matrices. Numerical experiments indicate that finite-element

models of non-self-adjoint problems are, generally speaking,

more sensitive to round-off errors than those of self-adjoint

problems. Convergence, when it exists, is not monotonic,

and specific information on rates-of-convergence does not

appear to be available. In short, the non-self-adjoint pro-

blem has a number of complicating features that have stood

in the way of· the development of a complete theory of asso-

ciated finite-element/Galerkin approximations.

In the present paper, we describe a fairly general theory

of finite-element approximation of a class of linear mixed

boundary value of the form

Au = f in n(2.1)

g2 on an2

where n is an open bounded domain in n-dimensional space

~n, an is its boundary, and A is any linear operator repre-

4

sentable as the composition of two linear operators Sand

T:

A = ST (2.2)

The boundary conditions may be mixed; an is the union of two

mutually disjoint sections, anI and an2, and the boundary

data is given on each ani in terms of operators BI and B2T

corresponding to A in some way that will allow the problem

to have a unique solution for reasonable choices of f and gi'

We shall cite some specific classes of problems subsequently.

The form (2.2) is, of course, quite general. By taking

S = I and T = a/ax (n = 2), we obtain a general class of

first-order boundary value problems. By setting S = T*, -T*

being the adjoint of T, we obtain the general self-adjoint

problem in which A = T*T. What is significant about the form

(2.2) is that it can be decomposed into a pair of lower-order

problems, which suggest so-called mixed Galerkin approxima-

-tions; e.g. Au = f is equivalent to

Tu = v(2.3)

Sy = f

By considering a general decomposition of the type (2.3)

in which S is not necessarily the adjoint of T, we uncover

a technique for studying the convergence and accuracy of fi-

nite element approximations which has more important ramifi-

cations than merely the study of non-self-adjoint problems.

5

We are referring to approximations based on multiple dec om-

position of a given problem. Such decompositions have aJ.-

ways been a popular thing to do in mechanics. For example,

we often choose to study, instead of Navier's equations in

elasticity,

the associated system of equations

u(k,m) - , i "}anE J E:km

_ iJ'- a ,

Or, instead of the Bernouli-Euler beam equation

4D u = p

we consider any of the systems

Du = e D2u = M Du = e

De = M D2r.1= p D2e = V

DM = V DV = P

DV = P

etc; or in the longitudinal motion of an elastic rod we may

replace

by the system

6

auIT = v

oVat = a

auax = e:

Ee: = 0

aoax = p (a - f)

etc.

In finite-element-Galerkin approximations of such sys-

tems, exactly how does the error in one component of the

solution influence that of the others? Does, in fact, decom-

posing the system make it possible to always obtain more ac-

curate results for various derivatives of the primal depen-

dent variable u? Does it pay to use higher order polynomials

for displacements than stresses, as some have suggested, or

is the converse true? What can be said about the numerical

stability of such mixed schemes? We shall attempt to provide

some answers for these and related questions for a class of

linear boundary-value problems in the developments to follow.

3. Prolongations, Projections, and Interpolations

Since Galerkin methods are based on the idea of projec-

tions, it is fitting that we record a few properties of ortho-

gonal projections of Hilbert spaces onto finite-dimensional

subspaces. The present section is devoted to a brief account

of the ideas of biorthogonal bases described in [3,4] and to

7

a review and slight extension of Aubin's work on prolonga-

tions [5J.

3.1 Some Notation. Most of what we have to say pertains to

projections of elements in a general Hilbert space H, but

in applications we generally have in mind the Sobolev spaces

Hm(n), defined as the completion of the space Coo(n) of func-

tions infinitely differentiable on a bounded domain n in Rn,

in the norm

2 J ~ a 2Ilulim = L.J(D-U) dx

n Ic:l.9Tl(3·1)

where dx = dxIdx2···dxn and we have used multi-index notation;

i.e., g is an ordered n-tuple of non-negative integers, g =

Ig Ia a 1 a2 an

= ar + a2 + ... + an; X =x X ···XI 2 n -(3.2)

a1 a2 an algiD

g a a a=-. - ... --a a a a a aax I ax 2 ax n ax I ax 2 ••• ax n

1 2 n I 2 n

It can be shown that Hm(n) coincides with the space of func-

tions with weak (generalized) derivatives of order ~ m; it

is also a Hilbert space, with an inner product given by

(3.3)

u

The dual of Hm(n) is denoted H-m(n), and HO(n) = L2(n) is

called the pivot space, since it is self dual and

2 I 0 -I"'CHCn)CHCn)CH(n)CH (n)c··· (3.4)

<Xl

We also remark that the completion of the space C (n) of in-ofinitely differentiable functions with compact support in n

in the norm (3.l)·isdenoted ~(n); also, the semi-norm

is also useful in describing properties of functions in the

quotient space Hm(n)/pm-l(n), where pm-I(n) is the space of

polynomials of degree ~ m - 1 on n.

3.2 Restriction Operators. The most obvious thing about fi-

nite element or finite-difference approximations is that they

_involve fully discrete representations of functions; that is,

they involve replacing a differential equation by an alge-

braic one in which the values of the dependent variable or

its derivatives of some order at a finite number of points

are to be calculated. In other words, the approximate pro-

blem is solved in a space of finite dimension.

Let R~ denote a subset of G-dimensional euclidean space.

G(the subscript h is to imply that elements in Rh may depend

upon some mesh parameter h such as the diameter of the lar-

gest finite element). Then the mapping Rh which sends any

element u in Hm(n) into a set of G real numbers in R~ is

called a restriction of Hm(n) to R~;

RGu = (ul u2 ••• uG) = uh '"

There are many ways to construct such restrictions, but we

shall consider a class of restrictions suggested by finite

element methods. We begin by representing n as the union

of E closed subdomains (finite elements) lie (lieis the

closure of ne) such that ne n nf = ~ for e ~ f, and we de-

fine locally over each ne a set of basis functions w~e)(~)

which are generally polynomials. Upon connecting all ele-

ments together to form a finite element model of n, we si-

multaneously construct a system of global basis functions.

This procedure is well known and is described fully in, for

example, [3]. For approximations in Hm(n) we shall gener-

ally take

k~m (3.8)

where, again pk(n) is the space of polynomials of degree

~ k on n.

The set of G linearly independent functions

form the basis of a G-dimensional subspace Mh of In'

general, the set {~i} is not orthogonal with respect to any

inner product. However, ~ variety of biorthogonal basis can

be constructed as follows:

J..U

1. For any choice of a Sobolev inner-product (3.3)

of order ~ m, construct the Gram matrix

i,j = 1,2,"',G

2. Compute the corresponding inverse,

Gij = (G ( s ) )-I(s) - ij

(3.10)

3. Construct the conjugate basis functions of order s

(Cf [1,2]),G

=2:i=l

(3.11)

4. The conjugate (or dual) basis functions so con~

structed have the property that

i j = 1 2 ... G~ "~ (3.12)

5. Finally, the s-th restriction of u E Hm(n) to ~~

is defined by

R (s)h u-

\'lherein

(3.13)

i iu (s) =- (u ,I{J (s) ) s

For example, the L2-restriction is simply

(3.14)

R(o)u = R u = JU~j( )dx =h - h 0

n

3.3 Prolongations and Projections. Now every function

U(~) in the subspace Mh is, by definition, a l~near com-

bination of the functions CPi(~); i.e., each U (~) E Mh is

of the form

U(~)

G

= "'"" ai

CPo (x)LJ ).-i=l

(3.15)

GPh from lRh

of R~ into

It is important to realize that the coefficients ai

Gelements of Rh' Thus, (3.15) defines a mapping

onto Mh. This mapping is called a prolongation

Hm(n), and we write

GI 2 G ~ i

Ph(u ,u ,'" ,u ) = L.-J u CPi (~)

i=l

are

(3.16)

Clearly, the prolongation Ph is defined by a given choiceG

of basis functions {CPi(~)}i=l' but it maps different re-

strictions {uts)l into different elements U(~) in Mh.

The compositions

(3.17)

mdefine projections of H (n) onto Mh. Indeed,

nCs)h U

G

= L (u,IP~S))S.pi(:~)

i=l

(3.18)

For example, the L2-projection of ~(n) onto M is

nCo)h u

It is natural to seek optimal prolongations or pro-

jections for a given restriction (see, for example, [5]).

Seek, for example, the restriction ui such thatr

iII u - Phur II s = inf II u - u II sUEMh

The answer to this problem is, of course,

i= u s (3.20)

Since (3.11) holds and

a-raUk

II u - U II = 0 = -2 (u .p) + 2 ~ u~G(s)s ' k s ~ J jk

j=l

Moreover, u -

inner product

orthogonal to Mh with respect to the

3.4 Finite Element Interpolation. It is well known that

the finite element method has to do with a special but very

useful method for constructing the basis functions lPi of

13

(3.7) or (3.8). In general, global functions are generated

according to the formula,

E Ne

'Pi (:~) = u 2:::e=l n=l

(e)Nn.l

a(e)t/; ~ (~) (3.21)

wherein E is the number of finite elements, Ne the number

(e)of nodes of element n , n N is the Boolean transformatione

. . aCe)matrlx correspondlng to ne, and ~N (~) are local inter-

polation functions, usually polynomials, such that

(3.22)

Nwhere ~e is a mode of ne and I~I ,I@I ~ m,m = integer (see

a[3] for additional details). By relabelling the 'Pi of (3.21)

we obtain a set {"'i}~=l (3.8); they have com2act support in

n, but their'corresponding conjugates 'P~s) do not.

We shall assume that the following conditions hold:

1. n is an open bounded domain in ~n that satisfies

the cone condition [6J;

2. let u(~) E: Hj (n),

a G-dimensional space Mh C

and let {~}G be a basis ofi i=l

k+lH (n), the elements of which

are piecewise polynomials of order ~ k, where j ~ k + 1 and

k > n/2;

3. let IThbe a bounded linear operator from Hj(n)

into HmCn), 0 ~ m ~ k + 1, such that

14

4. let the dimension G of Mh be increased by regular

refinements of finite element meshes [3]; then, it can

be shown [7] that as h tends to zero,

where c is a constant independent of h, lulk+l is defined

in (2.5), and Eu is the interpolation error

Eu = u - IIhU (3.24)

Interpolation error estimates such as (3.23) are fundamental

to finite element convergence theory.

4. Mixed Galerkin Approximations

We now return to the general non-self adjoint boundary-

value problem

STu = f in n(4.1 )

which we split into the equivalent pair of equations

Tu = v

Sy = f

in n,

in n,

Here ST = A is a continuous mapping of a Hilbert space U

into a larger space F which contains f (generally U = Hm(Q),

F = HO(n) = L (n)). The operator T is a linear mapping of2

U into a Hilbert space ~ which is also contained in a larger

nspace Q (often Q = F or ITF) and S maps ~ into F.

Definition 4.1. Let M~ and N; be G-dimensional and

H-dimensional subspaces of U and ~, respectively, spanned

by functions {~i}i=l and {~I}I=l generated using the fi-

nite-element method. Let (ffi~, Ph' Rh) and (~:' Pi' Ri

)

be the corresponding finite element approximation of the

spaces U and ~; i.e.,

GU ~ ~h

Ri:H = '>::!i V ':! E V~ ~ lRi , Ri':!

H H HPi: Ri -..~i , Pi "!i = Y! E ~i c V (4.4)

then the discrete, mixed finite-element approximation of

.(4.2) is

(4.5)

where R etc. indicate extensions of the operatorsi,ani

Ri,Ph,etc. to an.

Upon introducing the projections

(4.6)

.10

we see that

IT.e?U = V

IThSV = IThf

where

U = Phuh and y = P.e.v.e

(4 .7)

(4.8)

Following the nomenclature of Aubin [5], the system (4.5)

is called the exterior approximation of (4.2) while (4.7)

is called the interior approximation of (4.2) (since U and

yare in U and V). Upon elimination of V from (4.7), we

obtain

from which we solve for the finite-element approximation U

of the solution of (4.1).

Definition 4.2. The discrete operator Ah : Rg ~ R~given by

(4.10)

is called the stiffness matrix of the mixed finite element

approximation (4.5), and the vector

(4.11) .

is called the consistent load vector.

Again the external approximation of (4.1) is

(4.12)

17

while the internal approximation is

(4.13)

where

Definition 4.3. The functions

(4.14)

e = v-v - v (4.15)

where (u,v) is the solution of (4.2) and (U,V) is the solu-- -tion or (4.7), are called the approximation errors of the

mixed finite-element/Galerkin scheme.

It is a common feature of Galerkin methods that the

approximate problem can be posed in a larger space than that

containing the exact solution. For example, in the problem

where f E HO(O,a) = L2(O,a), for reasonable choices of boun-

dary conditions and data u E H3(O,a). In the decomposition,

2D u = v

we may actually consider the pair

Dv = f (4.16)

(4.17)

for appropriate choices of weights ~ and ~ and homogeneous

boundary conditions. Then approximations to u and v can be

taken from HICO,a) (or possibly HICO,a)). This is perfectlyo

..LV

admissable since H3(0,a) C H2(0,a) C HI (O,a), but it greatly

influences the type of convergence we obtain.-

With these observations in mind, we denote by U some

larger space, containing the "solution space rr U, into vlhich

the operator A is naturally extended by the decomposition

and the variational formulation of the problem:

u C u (4.18)

Naturally, we must also consider the possibility that U = U.

Definition 4.4. The mixed finite-element scheme (4.7)

is said to be convergent in energy if

lim II eu 11-h ,.e~0 u

where U C U.

= 0, lim II S~v II = 0h ,.e~O F

(4.19)

lim II Ew II - = 0h~O U

and lim II St:wII = 0.t~0 F

(4.20)

and if there exists a constant Mo > 0, independent of hand

.e., such that

(4.21)

for every wE U and W E ~,where Ew and ~w are the interpo-

lation errors [see (3.24)J

E = w - n w-vi - i -

19

(4.22)

Definition 4.6. The mixed finite-element approximation

of (4.2) given by (4.5) shall be referred to as stable if

there exists a constant M1 > 0, independent of hand i, such

that

(4.23)

or, equivalently, (4.7) is stable relative to U, U c ii, if

(4.24)

v wE U, wherein Ahi and Qh are given by (4.10) and (4.14)

respectively.

It often happens that the operator S in (4.1) has a

bounded inverse defined on its range. Then a stronger type

of stability can be obtained. Recalling that T maps U into

V and S maps V into f, we observe that the variational pro-

blem can be posed in larger spaces U ~ U and V ~ V (often

V = ii, as in the example (4.17)). It then makes sense to

introduce an addition~l definition:

Definition 4.7. The mixed finite element scheme (4.7)-shall be referred to as strongly stable relative to V ~ V if

it is stable relative to U and if a constant NI > ° exists

such that

(4.25)

v ~ EO.

The stage is now set for a look at convergence theory.

5. Convergence and Accuracy

The notions of consistency and stability of difference

schemes for linear differential equations are well known,

and there relationship to convergence is summed up in the

equivalence theorm of Lax. That our definitions given in

the previous section lead to the same sort of equivalence

theorem for finite element approximations is established

in the following theorem.

Theorem 5.1. Let the mixed finite-element scheme (4.5)

be stable and consistent relative to U ~ U. Then it is con-

vergent in energy.

Proof: Suppose that u is the exact solution to (4.1)

and TIhU is its orthogonal projection in M~. Since the scheme

is stable, we have from (4.24)

II U - TIhUIIu 2.' M~ 1 II TIhSTITU - IThSTItT IThU II F

= M~l I IIThf - TIhSTIt(TTIhU- Tu + y)\ IF

= M~IIITIhSY - TIhS(TIty)

~ M~ I II TIh S (y - ITty) I I F + M~ I I IQht (u - IThU ) I 1F

Since I 1IThWI IF ~ I Iwl IF and the scheme is consistent, we

can introduce (4.21) to obtain

However,

II eu I I ~ = II u - u II -u u

= II Eu II iJ + II u - IIhU II ii

Hence,

(5.1)

Since I IEul 10 and I Is~vl IF vanish as hand .e.tend to zero for

consistent schemes, so does Ileul 'ii' and the first number of

(4.13) holds. To establish that the second member of (4.19)

also holds, observe that

I I S~v I I F = II Sy - Sy II F

= IISy - SIl.e.TU- SIltY + SIl.e.TuIIF

= IIS(y - Il.e.Y)+ SIl.e.T(U- u)IIF

< II S~v II F + II SII.tTeu II F

< I I SJl:vII F + Mo II eu I I U

Therefore,M M

I IS~v I I F ~ (1 + M~) I I SJl:vIIF + Mo (1 + M~ ) I IEu II ii

which, in view of (4.20), completes the proof.

Specific rates of convergence are established in the

following theorem.

Theorem 5.2. Let the conditions of Theorem 5.2 hold,

G - H -and let the subspaces Mh C Ul Mt C ~ be generated using

piecewise polynomial finite element approximations of de-

gree k and r, respectively (that is, M~ C Pk(Q) and

Mf C Pr(Q)). "Let S, T, and the data be such that U =

and II Sy II F ~ II y II p ~ c II Sy II F' p 2. m. Then

I Ie I I ~ c hk+l-mlu\ + c lr+l-Plv!u m 0 k+l I - r+l

I I~vl Ip ~ C21r

+l

-PIY\r+l + C3hk+l-mlulk+l

(5·3)

(5.4)

wherein ci' i=0,1,2,3, are constants independent of hand t.

Proof: The proof follows immediately from (5.1) and

(5.2) upon introducing the interpolation error estimates

(3.23).

Estimate (5.4) is generally pessimistic, and we can

improve it for strongly stable schemes.

Theorem 5.3. Consider a mixed finite-element scheme

which is consistent relative to U and strongly stable. Then

(5.1) holds, but the error in v is such that

C,J

Proof: The fact that (5.1) still holds is obvious from

the definition of strong stability and the first steps in

the proof of Theorem 5.1. We now abandon (5.2) and use

(4.25):

II €v I I V ~ I I Ji:v I I V + II y - II.ty II ~

< I Ifv I I ~ + ~ I I I IThSII.t(Y - IT.ty) I I F

< II ~v I I _ + ~ II IThS (y - IT.tY ) IIV I F

/.1~ II ~v II V + fJ711S~v II F

The next result is obvious:

Corollary 5.3.1. Let S map ~ continuously into F.

Then for strongly stable, consistent schemes, (5.1) holds

and

(5.6)

where No and NI are constants.

Finally, we have:

Corollary 5.3.2. Consider a consistent, strongly stable

mixed finite-element scheme for which (5.6) holds. Suppose

that the polynomial subspaces M~ and M; of Theorem 5.2 are'

used to construct the bases. Then

c.."

lie II ~ c hk+l-mlul + c .e.r+l-Plv\u m 0 k+ 1 1 ~ r+ 1

I I~vl Ip ~ c2tr+l-P/ylr+l

where co'

hand t.c1' and c2 are positive constants independent of

Thus the error in y is, in a sense, independent of that

in u. We shall interpret these results in a later section.

6. Multiple Decompositions

The general procedure used to study mixed Galerkin ap-

proximations of the operator ST can be extended to multiple

decompositions of the type described in (1.5). We shall

outline the process below.

We begin by considering a general linear operator equa-

tion of the type

Au = f

where A is the composition of N + 1 linear operators,

(6.1 )

N

=Li=O

AN-i (6.2)

The operator A maps a Hilbert space U continuously onto a

space F containing the data f, and each component A of Ai

maps an intermediate space Vi continuously into Vi+l

; i.e.,

(6.3)

25

and there exist constants Mi,j such that

(6.4)

i=O,l,···,N. This also leads to inequalities of the type

(6.5)

etc., since the composition of continuous operators is also

continuous. Examples of such situations are plentiful; if

u(x) H4(0,L) and Au = b4u ~ h4u/d~4 (as in a beam problem

with a loading function p(x) in L2(0,L)), A maps H (O,L)

0continuously into H (O,L) but D = d/dx also maps H (O,L)

continuously into H3(0,L) and H3(0,L) contirluously into

2H (O,L), etc.

We next consider the decomposition of (6.1) into N + 1

component equations,

, i=0,1,2,'" ,N-l

(6.6)

The analysis of this collection of equations by finite-

element Galerkin methods proceeds as follows:

1. Let vi E Vi in (6.6), and identify projection oper-

ators llhi

of Vi onto finite-dimensional subspaces Mhi

. The

mixed Galerkin approximation (internal) of (6.6) is then

c'b

i=0,1,2,'" ,N

(6.7)

with Vo = u.

2. The approximation errors e and the interpolationv.1

errors Ev ' defined byi

ev. = vi - Vi1

and (6.8)

are easily seen to obey the "orthogonality" conditions

, i=O,l, .... ,N-l

3. The approximation U of u is determined by the equa-

tion

(6.10)

4. The mixed finite-element/Galerkin approximation

(6.7) is termed consistent relative to U, VI "',F whenever

(6.11)

vanish as ho,hl,'" ,hN tend to zero and when there exist

constants Mo,Mij, independent of ho,hl , ... ,hN such that

(6.12)

5. The mixed scheme is stable relative to U if a con-

stant C, independent of ho,hl," ',hN, exists such that

(6.13)

v wE U.

6. For stable schemes, a priori error bounds for the

error in u are obtained via (6.13) and the triangle inequa-

~ liE 11- + II u - nh u II U-u U· 0

IIEull- + lil-nh AN' "Ao(u - TIh u)LI coo

(6.14)

If the scheme is also consistent, we have

(6.15

7. Using the conditions (6.8) and (6.12), we find

the following error bounds for Vi' i=1,2,' ",N:

~ I \AiEv I I + Mi,i-ll lev·_ll Iv.i Vi+l l l-l

where the reccurance formula begins with the inequality

MI 0 (1 + Mo) II Eu I I + I IA IEv I I, cUI V2

N-l I+ ~ MN,N-il IEvN_i2' V

N-i

1=1

(6.17)

etc.

It is clear that stable and consistent schemes are,

once again, convergent.

8. Estimates (6.16) and (6.17) are usually pessimis-

tic, because they do not account for the fact that the

29

component operators Ai have bounded inverses defined on

their ranges. To take advantage of this, we shall term

the mixed scheme strongly stable if constants Ni . exist,J

VN_" i=O,1,2,' ",N, ITh = ITh ' VN+l = F. We1 N+l 0

then have

II eVN II VN

II II -eVN_i

VN

_i

< (1 + MN+1- i ,N- i) I IE I I _NN-i,N+l-i vN_i VN_i

(6.l9a)

i-l M+ "" eN, N- i ...j II Ev ' II -

~ ijNN_i+j,N N-l+j VN_i+jj=O

(6.l9b)

i=1,2,' ",N, where cij' Mij, Nij are constants independent

of the mesh parameters hi' ho'

Theorem 6.1. For strongly stable, consistent, mixed

finite element schemes for multiple decompositions of the

form (6.6) in which (6.4) and (6.18) hold, the error bounds

(6.19) hold.

This is one of the principal results of this study.

We shall interpret it in connection with some simple exam-

ples in the remaining sections.

.JU

7. Examples

Example 7.1. As a simple one-dimensional example, con-

sider the third-order problem

o < x < a(7 .1)

u ( 0) = U II ( 0) = u" (a) = a

We decompose the problem into three components,

dudx = v

dvdx = W

dwdx = f

u(O) = 0

w(O) = w(a) = 0

(7 .2)

Next, we construct finite-element approximations (U,V,W)

of (u,v,w). If the interpolation functions used in the ap-

proximations U, V, and Ware denoted ~i' ~j' and ~ respec-

.tively, the Galerkin approximations are determined from the

system of equations

a

~.(dU _ V)dx = 0;l. dx

o

a a

dV J dUn~(- - W)dx = O' tpk(- - f)dx = 0u dx ' dx

o 0

ic12 ... K' j=l 2 ... L' k=l 2 ... H where" " " " " "

and

L

W = Lj

31

wherein h = a/H, k = a/K, t = aiL.

The internal discrete problem leads to the equation

(7.3)

Assuming fE HO(O,a), we have TIh:H3(O,a) + Mh' TIk:H2(O,a) + Mk'

and TIt:HICO,a) + M. A pessimistic definition of stability

for the discrete model is to require a constant c > 0 to

exist

where

such that I IrrhDTItDTIkDTIhWI10 ?. c11TIhW113, V w E H3

(O,a),

IIu II :: IIu II m )' and the approximation is stablem H (o,a

whenever the interpolation errors Eu = u - IThU, Ev = v - ITkV,

Ew = W - ITtW are such that liEu 113, I IEv II 2' and IIEwll1 van-

ish as h, k, and t tend to zero and if constants Mij exi st

such that

etc.

< ~103 II u I I 3 '

<M31Iull,- I 3

With all of these preliminaries, it is now a simple

matter to obtain the following error bounds

M M MIleull < (1 + ~)IIEull + ~IIEvll + ~IIEwll3- c 3 C 2 C 1

M MI lev II 2 ~ M2 3 (l + ~ 3 ) I IEu II 3 + (1 + ~ 2 ) I IEv I I 2

M M+ 23 olliE II

c w I

oS.. t<11 3 (1 + M~I ) I IEU II 3 MI 3 f.102 I IE I I+ C v 2

M+ (l+~)IIE II

c 'vi I

(7.4)

The error estimates (7.4) portray a rather dismal pic-

ture of the mixed finite element model: the accuracy in each

variable is governed by the accuracy of the approximation of

u. In turn, U must be at least a cubic to converge, and

even then the accuracy is only of order h. Moreover, piece-

wise linear approximations, it appears, will not lead to a

convergent model at all!

What is wrong with this analysis is that in imposing our

rather weak notion of stability we have guaranteed only glo-

bal stability of the stiffness matrix while allowing its

components to behave as wildly as they please. In other

words, the approximation U depends continuously on f but ITkDU

may not depend continuously on V. This is unrealistic, since

it indicates that at least cubics are needed to get conver-

gence in H3(0,a). What we really want is convergence of

U, V, and W to u, v, and w in HI(O,a).

To achieve this, we redefine the notion of stability:

the scheme (7.3) is strongly stable if it is stable and if

there exist constants coi > 0, i = 1,2,3 such that

IIJIkDIlhUl1 ~ c0311JIhUIl , 'if u E H3o 1

IIn.tDITkvl10 ~ c0211ITkVlli , 'if v E H2 (7.5)

IIJIhDIl.twll~ cOlllJI.twll , V w E HIo I

33

If we denote by E = U - IThU (and E ,E ) the projectionU v w

errors, we see that for strongly stable schemes,

:£ -LII ITkDITh (U - IThU) IIC03 0

= ~ I IV + ITkD ( u - IThU) - ITk v I IC03

Now, using the triangle inequality (I Ie I I :£ I \E 1 I + I IE I I)u u u

II eu III

II ew III

< (1 + 2-) liE IIC03 u 1

:£ (1 + -L) I IEw I IICOl

+ 1 liE II02COI W I

(7 .6)

Suppose that we use polynomials of order i, j, and k

for U, V, and W respectively. Then use of (7.6) together

with (3.23) yields the error estimates

I Ie I I ~ K hilul + K kj lvl + K lklwlu I 00 i+l Ol j+l 03 k+l

I levi I ~ Kllkj IVI. + Kl2lklwl

1 J+I k+l

I lewl I ~ K22lk

lwlI k+l

This is a remarkable result. It says that the order

of error in w is independent of the degree of polynomials

used in approximating v and u, but that the error in u de-

pends upon the approximations of all components, u, v, and

w. If piecewise linear polynomials are used for w, it does

us no good to use any higher degree polynomials for u and

v. However, if (say) cubics are used for w, we should at

least use cubics for u and v to obtain the same order of

accuracy for each. If cubics are used for approximating w

and linear polynomials for u and v, the errors (in the HI -

norm) in ware of order h3 while those in v and u are only

of order h (assuming h = k = l).

Example 7.2. A special case is the self-adjoint problem

u(O) = u(a) = 0

which leads to the pair,

(7.8)

. For consistent strongly stable schemes, we have

II eu III(7 .9)

35

Thus, for this problem we can draw the same sort of con-

clusions we derived in the first example.

8. Some Concluding Comments

We can now record a number of important conclusions

of our investigation.

1. First, we make the general comment that our ap-

proach to a theory of convergence of mixed finite element

models has made use of concepts of consistency and stability

of Galerkin approximations. The idea of consistency amounts

to requiring that the coordinate (or interpolation functions

be complete in an appropriate norm, and that the Galerkin

approximation of a continuous operator be also continuous

for arbitrary choices of mesh parameters. The notion of

stability is interpreted in the sense that the approximate

solutions must depend continuously on the (discrete) data,

'and this implies that the matrices involved must have a

bounded inverse for arbitrary choices of the mesh parameters.

For mixed finite element models, this leads to alternate

definitions of stability since, in some cases, component

equations approximated in a certain decomposition may, them-

selves, have bounded inverses. When this happens, we say

that the scheme is strongly stable.

2. This approach is quite general. We obtain as spe-

cial cases the conventional convergence theory and error

estimates for "displacement" formulations. However, the

36

method also applies to non-self-adjoint problems and includes

the decomposition of self-adjoint operators as a special case.

3. For consistent and strongly stable schemes, the error

in the N-th decomposition variable is independent of all the

errors in the remaining approximations. For instance, in

approximating the decomposition

Du = v

Dv = \'1

Dw = f

of (8.1 )

the error in w is independent of the degree of polynomials

used in approximating u and v. However, the error in u does

depend on a linear combination of the interpolation errors

in u, v, and w. Consequently, if linear functions are used

to approximate w, it does no good to use any higher degree

polynomials for u and v. Likewise, the use of high degree

polynomials for u does not improve the accuracy in w. To

obtain the same orders of accuracy for u, v, and w in the

same norms, polynomials of the same degree and proportional

mesh sizes should be used for each approximation.

4. For mixed approximations which are not strongly

stable, the mixed method may often have little to offer.

Here all approximation errors are linear combinations of all

interpolation errors. This has advantages in problems in

which only a single decomposition into two operators of the

same order are used. Then equal orders of accuracy can be

obtained in both dependent variables in the same norm.

37

5. If Ahf

is any discrete approximation of an operator

A obtained using mixed finite element approximations, the

consistency and stability requirements are manifested in

respective inequalities I IAhfUhl IF ~ NI luhl IU' I IAhfuhl IF

> MI IUhl 1-. The errors then involve terms likeU

(I + M)l IEul lu' It is clear that N ~ Amax' M ~ Amin' where

A's are eigenvalues of Ahl; thus the coefficients in the error

estimates depend upon the condition number of the stiffness

matrix Ahf' If these coefficients are to be independent of

the mesh parameters hand f, then hand l cannot be allowed

to vanish independently. Generally, we must set

hI = \I = constant

or, in multiple decompositions, we set some combination of

m(hi/hj) or (hi/hj) equal to a constant. This done, the co-

.efficients behave as amplification factors of the errors.

6. For consistent, strongly stable schemes such as

(8.1), the first approximation u should be more sensitive

to round-off errors than v or w, since its error bound con-

tains products of the amplification factors of v and w.

For schemes which are not strongly stable, the opposite

situation occurs.

38

Acknowledgement. This work was done while I was a visitingprofessor at COPPE (Coordenacao dos Programas de Post-Gra-duacao de Engenharia) of the Universidade Federal do Rio deJaneiro, in Rio de Janeiro. My time there was made particu-larly pleasant by the hospitality of Messrs. Nelson Ebeckenand Cid Gestieiia. I am also indebted to my colleague, Dr.J. N. Reddy, with whom I have had many discussions of ma-terial related to this work. I also wish to acknowledge,with gratitude, that all of my recent studies on mixed fi-nite elements have been supported by the U. S. Air ForceOffice of Scientific Research under Contract AFOSR-74-2660.

References

1. Reddy, J. N. and Oden, J. T., lIMathematical theory

of mixed finite element approximations,lI Quarterly Appl'd.

Math. (to appear).

2. Oden, J. T. and Reddy, J. N., liOnmixed finite

element approximations," TICOM Report 74-2, Austin, 1974.

3. Oden, J. T., Finite Elements of Nonlinear Contir.ua,

McGraw-Hill, N. Y., 1972

4. Oden" J. T., lIGeneralized conjugate functions for

mixed finite element approximations of boundary-value pro-

blems," The Math. Foundations of F.E.M. Appl'ns. P.D.E.,

Academic Press, N. Y., 1973, pp. 629-669.

S. Aubin, J. P., Approximation of Elliptic Boundary-

Value Problems, Wiley-Interscience, N. Y., 1972.

6. SObolev, S. L., Applications of Functional Analysis

in Mathematical Physics, Am. Math. Soc. Trans., Providence,

R.I., 1963.

7. Ciarlet, P. G. and Raviart, P. A., "General Lagrange

and Her~ite Interpolation in ~n with Applications to the

Finite Element Method," Arch. Rat. Mech. Anal., 46, 1972,

pp. 177-199.