.1

A STABLE SECOND-ORDER ACCURATE, FINITE ELEMENT SCHEME FOR THEANALYSIS OF TWO-DIMENSIONAL INCOMPRESSIBLE VISCOUS FLOWS

CoV- - T~J. T. Oden and O. Jafluotte _,:..,C 01.1 u T t

The Texas Institute for Computational MechanicsDepartment of Aerospace Engineering and Engineering

Mechanics, The University of Texas at Austin

1. INTRODUCTION

For the last several years (indeed, dating back to the Second Confer-ence on Finite Elements for Flow Problems in 1976), the use of exteriorpenalty methods and reduced integration as devices for producing veryefficient finite element methods for the analysis of incompressible viscousflows has been advocated by numerous investigators. The idea is to useexterior penalty methods to handle the incompressibility constraint,div u = 0 , and then to use reduced integration of the penalty terms to"unl~ck" the system (which basically frees the dependence of the penaltyparameter £ on the mesh size h .

In 1980, however, we were able to show both mathematically and experi-mentally that many of the favorite RIP (reduced-integration-penalty) schemesare numerically unstable. These difficulties were discussed in Oden'slecture at the Bnaff Conference in 1980 and are summarized in the finalconference paper by Odenl, Complete d~tails of this analysis are given inthe papers of Oden, Kikuchi, and Song2,3 together with a more complete listof references on this subject.

From the mathematical point of view, the key to the stability of thesemethods is the so-called LBB-condition, to be given in the next section.The once widely used Q2-elements for velocities (tensor products of quadra-tures) with 2 x 2 Gaussian integration (or, equivalently, with discontinuousbilinear pressures) and the well-known Ql-elements (bilinear) with I-pointintegration of the penalty terms (or, equivalently, constant pressures)foil the LBB-condition for most choices of boundary conditions in that theylead to an LBB-parameter ah which depends upon the mesh size h. If oneuses Q2 elements for velocities but only I-point integration, then the LBBcondition is satisfied with ah independent of h, but the rate-of-conver-gence of the method is suboptimal. The question naturally arises as towhether or not it is possible to produce a stable RIP method which exhibitsan optimal rate of convergence in the "energy" (Hl_) norm.

At the third symposium at Bnaff, Oden announced two types of elementswhich were candidates for such optimal RIP methods: the Q2-element forvelocities with discontinuous piecewise linear pressures (referred to hereas the Q2!Pl-element) and the 8-node isoparametric element with discontin-uous piecewise linear pressures (here the 18/Pl-element). These were sub-sequently coded and by July 1980 were producing encouraging numericalresults (see Oden 4 ). Similar numerical results were obtained by otherinvestigators in the intervening months (see Zienkiewicz and TaylorS andEngleman and Sani 6) .

1

In the present paper, we examine these two elements in greater detail.First, we show that the Q2/Pl-element does, in fact, satisfy the LBB-conti-tion with ah independent of h whereas the 18/Pl element apparently doesnot, except under special circumstances. We also attempt to provide resultsof numerical experiments which support some of our theoretical findings.

Our main result is that the Q2/Pl-element is stable and exhibits anoptimal convergence rate in the HLnormo \o.lhile18/Pi is "millilJ" uRstab' e.We also suggest certain ways to stabilize this element for certain classesof problems.

2. STATEMENT OF THE PROBLEM

It suffices to consider the following two-dimensional Stokes problem:

vlm + 'Vp

div u

f in 0

o in 0 (1)

u = 0 on ao

Here V is the viscosity, ~ the velocity field, p the hydrostatic pres-2sure, f the body force, and 0 is an open bounded polygonal domain in ~To rew~ite (1) in a variational form, we introduce the notation

v

II • 110 = the inner produce and norm on H

L (u ,v ) - inl~i,j~2 i,j i,j 0 - nerproduct on V

bilinear form on V

K {v E V I div v O} .

Then we consider the problem of finding (~,p) E V x H , fop dx = 0 ,such that

2

In the present paper, we examine these two elements in greater detail.First, we show that the Q2/P1-element does, in fact, satisfy the LBB-conti-tion with ah independent of h whereas the I8/Pl element apparently doesnot, except under special circumstances. We also attempt to provide resultsof numerical experiments which support some of our theoretical findings.

Our main result is that the Q2/P1-element is stable and exhibits anoptimal convergence rate in the Hl-normowhile 18/Pi is "millily" vRstab] e.We also suggest certain ways to stabilize this element for certain classesof problems.

2. STATEMENT OF THE PROBLEM

It suffices to consider the following two-dimensional Stokes problem:

v~~ + 'Vp = f in 0

div u o in n (l)

u = 0 on an

Here V is the viscosity, ~ the velocity field, p the hydrostatic pres-Zsure, f the body force, and 0 is an open bounded polygonal domain in ~To rewrite (1) in a variational form, we introduce the notation

v

11'110 = the inner produce and norm on H

L (u v )l~i,j~2 i,j' i,j 0

innerproduct on V

a(u,v) V(~'~)l= bilinear form on V

linear form on V

K {v E V I div v o} •

Then we consider the problem of finding (~,p) E V x H , fop dx = 0 ,such that

2

a(u,v) - (p, div v)

(q, div u)

f(v)

o

¥v~V

¥qEH} (2)

Of course, any smooth solution of (l) satisfies (2) and any solution of (2)is a solution of (1) if interpreted in an appropriate weak sense.

A regularization of (2) can be obtained by introducing a penalty par-ameter E > 0 and seeking u E V such that_E

a(u ,v) + l (div u , div v)-E - E -E

f(v) ¥vEv (3)

from which one can calculate an approximate pressure

1 d'P = - - 1V UE £ _£

as

(4)

It is easily shown that there exists a unique solution ~£ to (3) for any£ > 0 and that u£ -+ u in V and PE -+ P in H as £ -+ 0 ,where (u,p)is a solution of (2) (see, e.g. Oden, Kikuchi, and Song 3)., -

For the approximation of (3), we first consider the case in which nis a rectangle on which a uniform mesh of E rectangular Q2 elements ne hasbeen constructed, We introduce the discrete spaces

Vh = {vh = h h I(vl,v2)

v~ln Ee

h 0 -v. E C (n) ,1

1 < e < E , i = l,2} (S)

We then seek

1 < e < E} (6)

( h h d' h)q ,£p + 1V u 0E _Eo ) ( 7)

As shown by Oden and Kikuchil, RIP schemes are numerically stable when-ever the following discrete LBB-condition is satisfied for ~ independent

3

of h There exists ~ > 0 such that

h hI(q , div ~ )

lI~hlll

(8)

where Bh is the adjoint of+ boundary conditions) andapproximating (qh, div ~h)O

the discrete constraint operator Bh (= divh1(',') is the numerical quadrature operator

h h(q , div v ) ::::

B*h

3. MAJOR RESULTS

(9)

1·1-L2(n)

The major results of our study are summarized in this section. Fulldetails and the results of numerical experiments will be given in theexpanded version of the conference paper.

Theorem l, For the Q2/Pl scheme described earlier, ker Bh = ker B* = ~and there exists a constant, a, independent of h , such that V. qh E ~ '

f h hn q div ~ dx

D

The pressure of the center node in the Q2-element is essential for the proofof this result. Without it, our results indicate the method is unstable.

T553X 2 EOSmtb ~8 I__ i e.. yP ·cReme £01 tcble lii .• , .:a -------.~ -.!s-genA ......,u •oMI.: 1:m p m '(if,-, <1h =O(h):::aaa tlie-a~eme-'(J

However, we have_~l~0_es~ap1ished the following results: . 1 ~- -~~n~.==:::gul-aLmesh of I81PI=crl~m~t~;-4:f-~~y~~~~'~-~le~~~~~,replaeed-hy--a-·(hlPl-element,--the-reeult-ing---scheme-is-stable...(see-.~Fig-.,-21-;··---··!--1.

~ As is the case with many RIP methods, the "unstable" 18/Pl-schemescan perform well in the case of smooth solutions on a regular mesh on asmooth domain.

)~ The rates of convergence of Q2!P1-e1ement velocities in thenorm is 0(h2)- which is optimal; likewise, the pressures converge inat a rate of 0(p2) .

4

C{ -\ ...'V '1.-\.,"\ ~.,.MIc..'

PERTU~f3£o

STABLE .,4!1./D tJNST~I3l.E RIP/"LA Q flA-AJG/ A-JU ~ £,7Ji 01;) j AJ r1...

~~ABILITY OF gOK~7wo - D I ~E.'" S/ QUk¥G V tr CQ{/f

~p FINIT~ ~~&Wi~TMETHODS

}CUJltJ PIU~~EU~iQR gTOl€ESIAN FLOWS

J. Tinsley aden

and

Olivier Jacquotte

Texas Institute for Computational MechanicsDepartment of Aerospace Engineering and Engineering

Mechanics, The University of Texas

STABILITY ~ ~ FINITE

J. Tinsley Oden

and

Olivier Jacquotte

the pressure field p, t~e latter/

/with the inc~pressiblity

s based,~uch f~rmul

~ the mid-19AfOs,however, much

ressib1e, Stokesian fluids, two

l. INTRODUCTION

Iaso-cal1ed primitive variable formulations of problems

the velocity

introduced over a decade

interest has arisen in a re1a~/family o~finite element methods developed

around the ideas of hand~~the incompressibi~ constraint by means of

an exterior penalty~niqUe and then under-integra~ the penalty terms

to free the depenft6nceof the penalty parameter £ upon th~esh size h.

/One great advantage of such RIP (reduced-integration-penalty)

/element ~thods is that they reduce the number of unknown fields

/

one-~egularized velocity u ; the pressure can be computed a posteriori~~ • -E

as a post processing operation.

a Lagrange multiplier

div u = O. Finite element

Such RIP methods were developed and discussed by several authors,

and we mention in particular the works of Malkus [lJ,ll], Hughes [10], Malkus

1

2

and Hughes [l~, Reddy [2t], Bercovier [2], Engleman and San! [j1, and the

references therein. In 1980, however, mathematical analyses indicated that

some of the more popular RIP methods might be numerically unstable [1},l8,lg,20].

It was discovered that while certain of these methods perform well in problems

with smooth solutions for which regular uniform meshes are employed, serious

oscillations in the pressure approximation can occur when the data or the

mesh pattern are mildly irregular, and these oscillations increase in ampli-

tude as the mesh is refined.

Oden, Kikuchi, and Song [20] attributed the deficiency of these unstable

methods to their failure to satisfy a key stability criterion which they

referred to as the "LBB-condition,1l making reference to the work of

Ladyszhenskaya [11] on existence theorems of viscous flow problems and of

Babuska [l] and Brezzi [3] on the approximation of elliptic problems with

constraints. The discrete LBB-condition of Oden, ~ikuchi, and Song is

basically the requirement that the discrete approximation B~ of the

transpose B* of the constraint operator B::::div be bounded below as a

linear operator mapping the space of approximate pressures onto the dual

of the space of approximate velocities. For example, one form of this

condition is that there exist an ~ > 0 such that for all q E Qh *-n h '

< sup"h II qh II L2(Q) Iker B~ - Vh

Related conditions for mixed finite elements were discussed by Fortin [8]

and Girault and Raviart [-9]. The possibility of unstable pressure approxi-

mat ions of an RIP method is signalled by the existence of a parameter ~

* Definitions of terms displayed here are given in Section 2.

in the corresponding discrete LBB-condition.

O(h). We also

which dcpcnds upon the mesh size h. Indced, the fact that a mesh-dependent

~ corrcsponds to RIP methods with "spurious pressure modes" is supported

by the theoretical and numerical results of Oden et al [20] and by extensive

numerical experiments of Malkus 1l.]. Equally important, the behavior of

~ as a function of h governs the asymptotic rate of convergence of

RIP methods.

An important question that has arisen from these considerations is

whether or not stable RIP methods exist which converge at optimal rates in

the energy-and L2-nor~s. The present paper is directed at resolving this

question for a restricted class of problems by estimating the stability

parameter ~. -- --. ~ .. -, - _... _._. -. - .

S~e. 1\.e)v..eJ; ~~~~ ~ 4~aJ,~·e..:~ r---o.- ....el~ o....e ~

~ 0-. ~Q. 4~~ c..o1. 0(..f~'-\~ O"-L. ~ r P~ "....~ (0'1-~ stc)~' ~~~ ~ ~ 1+~~.

Among specific results we establish here is that the Q2/Pl (bi-

quadratic velocity/linear discontinuous pressure) element is stable with

~ = 0(1) while the 18/Pl (eight-node isoparametric for velocity/linear

discontinuous pressure) e1emen~ is unstable with ~

investigate the effect of averaging the pressures in the unstable elements

(pressure filtering). For the I8/PI\

for example, we find thatelement,

averaging restores the stability of the method and renders a method which-

converges at an optimal rate in L2 but which exhibits an accuracy regarded

as inferior to the Q2/P1-element. We also reexamine ah for the Ql/PO

(bilinear velocity/one point integration) and'find ~ = O(h). This

improves the estimate of 0(h2) given by Oden, Kikuchi, and Song [2D] and

agrees with the result of Carey and Krishnan [4] obtained independently.

3

2. STATE:-1ENT OF THE PRORLEM

Let n denote an open bounded region of IR2 with boundary an.We consider the two-dimensiona! Stokes problem an n, which involves finding

a velocity field u = (u1,uZ) and a pressure field p such that

-v6~ + ~p f in n

div u = 0 in n

u = 0 on an

(2.1)

..

,"

where v is the viscoscity of the fluid, (v = const. > 0), and f is the

body force, assumed to be a prescribed vector field with components

2f. (L (n).1 ,

..

.-.'

We recast (2.1) in a weaker variational framework by introducing the

spaces

5

and the forms

Q L2(n) (2.2)

a:V x V -+ m., f:V -+ m.

a(u,v)2

v(u,v)l' f(v) = L (f.,vi)- - - i=l 1

(2.3)

for all u, vEv, where (',')1 and (',-) are inner products on V and Q,

respectively, and are given by

(v,w)o = Invwdx ; v,wEQ

" (2.4)2 au. av.L 1 1 u,vEV(~,~)1 =.. (ax.' ax.) ;

1,J=l J ]

The partial derivatives in (2.4)2 are interpreted in a distributional sense.

We proceed by considering the problem of finding (u,p)EV x Q such that

a(u,v) - (p,div v) f(v) VvE V

(2.5)

(q,div u) o

It is easily verified that any solution of (2.1) satisfies (Z.S); any

solution of (2.5) satisfies equations of the form (2.1) in a distributional

sense. Under the conditions stated, it is also known that (2.5) possesses

a solution (u,p), with u uniquely determined by each choice of f and p

unique/up to an arbitrary constant.

Problem (1.1) can also be interpreted as the characterization of a

saddle point of the functional

L:V x Q -+ IR

6

L(v,q) ~a(~,~) - f(v) - (q,div v) (2.6)

with q clearly a Lagrange multiplier associated with the constraint,

div v o in Q.

An alternate formulation of problem (2.S) can be obtained using an

exterior penalty method. Let £ be an arbitrary positive number. By

seeking minimizers u E V of the penalized functional,-£

we are led to the variational problem

(2.7)

-la(~£, ~) + £ (div ~E:'divv) f(v) Vv E V (2.8)

This problem, which is uniquely solvable for u-£

for any £ > 0, involves

a single unknown field u •-£

Once u-£

is obtained, an approximation

of the pressure is obtained by the computation

-1-£ div u

-£(2.9)

The forms a(·,·) and fee) are continuous and a(·,·) is V-elliptic.

In addition, Ladyszhenskaya [li] has shown that a constant a > 0 exists

such that

a II q II 2 2 supL (n)/lR vE V-

(q,div v)

(2.l0)

7

where II~II = 'i(v,v) l' Under these conditions, the sequence {(u ,p )}-£ £ 0£ >

of solutions of (2.8) and (2.9) converge strongly in V x Q/lR to the

solution (~,p) of (2.S)

We remark that the penalized problem (2.8) can be interpreted in

another way. Returning to the saddle point problem for the functional L(',')

of (2.6), we introduce the perturbed Lagrangian

L :vxQ-+lR£

L (v,q)£ -

(2.ll)

for all qEQ, which represents a regularization of L(·,·) with respect

to the multipliers q.

are characterized by

For each £ > 0, saddle points (u ,p) of L(' .)-£ £ '

a(u ,v) - (p ,div v) = f(v)-£ - £

(£p + div u ,q) = 0£ -£

VvEv

(2.l2)

Upon solving the last equation in (2.12) for p , we obtain (2.9).£

Substi-

tuting this result into the first equation in (2.12) gives precisely the

penalty formulation (2.8).

3. FINITE ELEHENT APPROXUtATIONS

We shall outline briefly features of certain finite element approxi-

mations of (2.8). We confine our attention to cases in which n is

rectangular or is the union of rectangles and, for simplicity, to.uniform

meshes of rectangular elements of maximum length h. For a family of

such meshes with E = E(h) elemen~s, we introduce the discrete (finite-

dimensional) spaces,

~

Vh = {~h

Vhi In E Qk(ne); Vh. = 0 on ane 1

8 f

1 < e < E, i = 1.2} (3.1)

Qh = {q EL2(n) I qh Inh e

E P (n );r e

1 < e < E. r. > O} (3.2)

Here Qk(ne) is the space of tensor products of complete polynomials in

of degree < k defined on finite element TIe and P (0 )r e

is the space of complete polynomials of degree

Clearly, for every h,

< r defined on O.e

In addition to the spaces vh, we shall also consider cases in which Vh

is constructed using 18-elements:

9

18 eight-node isoparametric elements (3.3)

We also consider composite elements which employ both QZ and 18-subelements.

In any case, the finite-element approximation of the penalty formu-

equality (Z.8) for all

lation (2,8) consists of seeking

hv E V •

u£ E Vh which satisfies the variational-h

However, the performance of the resulting

method depends strongly on the relationship between £ and h (see Falk

r7]). For fixed mesh size h, the use of a small £ (say £-s10 for

most reasonable meshes) leads to a "locked" solution (which approaches

other hand, unlocks the problem in the sense that a nonzero solution

zero in V as £ tends to zero). The use of larger values of £, on the

£~h

can be obtained, but then the incompressibility constraint is not adequately

enforced. The solution to this paradox_ is to use reduced- integration for

evaluating the penalty terms in (2.8) (rl!,l!,l',~).

Let I('t') denote a numerical quadrature rule for integrating approxi-

mately the product of two element-wise continuous functions f and g:

I (f, g)

E Ge\ \ e e el l.. W,f(~.)g(~,)I . 1 J -J -Je= J=

(3.4)

Here G is the number of quadrature points in element e, W: are thee J

quadrature weights, and ~: are the quadrature points in element e.-J

By

choosing G sufficiently large, it is always possible to obtain the exacte

value of the integral InfgdX from I(f,g) for polynomial f and g-.

The RIP-finite elC'OIcntapproximation of (2.1) then consists of seeking

C E h~h Y such that

a(uh£,vh

) + c-lI(div u£,div vh

)- - -h-

10

v v E yh-h

( 3 ..5 )

For each choice of yh and quadrature rule 1('.·) there is defined

implicitly in the approximation (3.5) a space Qh of approximate pressure

Indeed. the specification of values of qh at each integration point within

an element in the definition of I(',') uniquely defines a piecewise

continuous polynomial over each element. For example. if Q2-elements are

used for the velocity approximation, a 2 x 2 Gaussian quadrature rule for

I(·,·) corresponds to a piecewise bilinear pressure.approxi~ation, etc.

In general, an m x m Guassian quadrature rule corresponds to a space Qh

of pressures which are discontinuous at inter~lement boundaries. but which

have restrictions in ~(ne) in each finite element.

Once (3.4) is solved for the velocity field

pressure approximation is computed as

~~, the corresponding

-1 £ e-£ div uh(~')

- -J

where (: are. again, the quadrature points in element n. Note that Qh-J e

is such that (3.6) uniquely determines

Under the conditions established thusfar. (3,S) is uniquely solvable

for £ for any £ > O. Ho\vever, the behavior of£ and pC h

~hu as £ or-h h

tend to zero depends upon more delicate features of the approximation.

*Let Bh and Bh denote the discrete operators,

B*hh' h'Q -..V

(3 ..7)

where [.,.J and <:.,.;> denote duality pairings on Q' x Q (Q = Q' = L2(Q»

and V' x V respectiv~ly (i.e., Bh and B~ are the discrete approximations of

div and -grad plus boundary conditions defined by 1(',-». Then, the

discrete LBB-condition for problem (3,5) (or (3.7» is as follows:

There exists a number ah > 0 such that

< supv E Vh-h

I(qh,div ~h)

II~hlll(3.8.)

The behavior of as h tends to zero and the structure of ker B~

governs the stability of RIP methods. In particular, let Eh(~'P) denote

the distance function

inf II u - vhlll + inf II p - qh" 2f Vh - - E Qoh L (n)

vh - qh

defined on V x~, ~ = {q E Q I fQqdx = oJ. Then one can show (see Oden

and Kikuchi [20]) that if (~JP) is the solution of (2.S) and

the solutions of (3.5), (3.6) in Vh x ~h,,

(3.10 )

II £11 -1 -2p - Ph L2(Q) ~ C(l + ~ + ~ )(Eh(~'P) + £)

where C is a generic constant independent of u, p) £, and h.

The remainder of this paper is devoted to the study of (3.8.) and

estimations of the stability parameter ~.

4, LBB CONDITIONS FOR CERTAIN RIP }lETIIODS

uJQ. .-Ja.h C2{)t.:~ 1Ale ~ak:.e:l:fJ ~~ oI~ ~ ~

~cuJ:e. L~B ~~ (3.1l) 8~~ (c«- ~f.f&w~ Rrp-meJf..~:1) Q.L/~ ~~[~~ 0.. ~lC~r~ ~ 0'-1 ~ ~VQ.~

of (1.3) o.-d (.2. \.!) ~ AN)ed lAjd~ ~(~~) veIoc..:teoa.-cle.:~c;I;o~ r~~'~ ~ w~~~~

~ ~ ~ I<~~ [t9]. ]2) I8/Pl elements [eight-node isoparametric elements for velocities)

linear pressures]

3) Ql/PO

elements [bilinear velocities, piecewise constant (one-point

integration) pressures]

4) Composite elements [elements consisting of two or more of the

above elements as subelements]

In all of these cases, we have*

(4.l)

even though I(div ~h,div ~h) f (div ~h,div ~).

I* This condition docs not hold for all RIP-methods; for instance, it is not

true for the Q2/PO-elcmcnts studied in (20].

-rC. ~ ...<..:,...e 4R4.b; ~~ ~ 6f,i' a..."I Aloe L M-o....<t<>..vl<X~ ~ $~ -<.~ A4 .pf1t,,,!~(f fJ.~.

~ I. LeI: ~~ (4,1.2.) e.Dfd ~ ~~~~- ~-~ yt a.-J (Q" k ~~-v.deo! A-0':""o- Q,L / P-j, - ~~~ ~ Br;.-'J-; ~'8~" ~ ~ ~t-aJo...~·t(J 1'~eA:~de. ~ ~. 'oU.,~ Lf3 (3 .. ~t:.~ (3. IJ) ~ C\.. 1'cP~t.ye..~kJ~~r~-4« ( d~= O(~))O .

Theorem II. Let conditions (4.12) hold and suppose that Vh and

Qh are defined ~ 18/Pl-elements. Then dim ker B~ = 3 and the stability

parameter ~ in the discrete LBB-condition (3.11) depends linearly on h:

~ = O(h) 0•• -- +- - -- -- .. -.-.- - ._---.-. '.'

- ~.' ..---.- .-. - - ..'- ..--... ---._- -- - ........ - -

Theorem tIT. Under the assumptions of Theorem II, if Vh and Qh are

defined using Ql/PO-elements (i.e., bilinear velocities and ~ ~-point

integration for the penalty), then dim ker B~ = 2. and ~ = O(h). 0--:-- ......

..---.-- +-.+_.---- -+ --- - •• --- -------.--.-

Theorem f{. Let conditions (4.12) hold and suppose thathV and

Qh are defined using composite 18/Pl - Q2/Pl-e1ements of the ~ shown

in Fig. 1· Then dim ker Bh = 1 and the stability parameter ~ appearing

in the discrete LBB-condition is ~ positive constant independent of h;

(~ = 0(1»).. ·0--- - -- ----.- .. -.

,~

S. f'..TU~IERICAL EXM1PLES

The results of several numerical experiments are described which are

designed to verify the theoretical results with regard to the Q2/P1'

I8/PI' and the composite elements described earlier. We consider an

L-shaped domain n partitioned into 64 square subdomains, as shown in

We will be inter-We take V = 333, and the penalty parameter

Fig. i. The fluid is subjected to a constant body force f = (0,-100).--5

£ = 10 .

ested in the computed-hydrostatic pressure across the section AA' defined

by: y = 0.80. Each subdomain corresponds to a finite element; the velocity

on each element is interpolated at 8 or 9 nodes and the pressure by its

value at 3 points. Thus, various choices of how to handle the ninth node

lead to meshes with 18/P1, Q2/PI or Composite/Pl elements. We

will be interested in ~reecases involving these elements:

Hesh 1: All the elements are Q2/Pl elements

Mesh 2: All the elements are 18/Pl elements

Mesh 3: Adding 16 centroid nodes, we obtain 16

composite elements as shown in Fig .. 1.

The results reported here were obtained using the FIDAP code for problems of

incompressible viscous flow [6]. •

, -

., •

Figures ~~ and fshow the comparison betwecn the results obtained

IS

with the Q2/PI element (Mesh 1) and thosc obtained with meshes 2 and 3.Fig .. ~ illustrates the major diffcrence betwccn the Q2/Pl and the

18/PI element: the former involves a pressure which seems to be smoothly

distributed along the section AA' while the latter yields a pressure

with scvere oscillations. We note, however, that the values of the pressure

obtained at the centroid of each element are close to the values obtained

withe the Q2/Pl element, which suggest that this unstable solution can be,

stabilized by a filtering operation which effectively uses these averaged

values of pressure.

It is also remarked that the oscillations seem to come from the

spuriou~ modes in ker B~. The smoothing device may be equivalent to an

a posteriori elimination of these spurious modes..;;:::s:: __ •. _ .....

Finally, the composite elements lead to a quite smooth solution as

indicated in Fig .. f, which is close to the.solution obtained with 9-node

elements, except that for this element h2= 0.2S, while for the Q2/Pl

2element h was equal to 0.0625.

, ,

We also note that when the body force f derives from a potential:

46

f = -Vv then the unique solution for the Stokes Problem is

u = 0

p = -v

In this example, f = (0,-100) and v = lOOy = -po

,The numerical results obtained by different methods are summarized in

the Table 1.

Table I: Norm-evaluation obtained by:

Method l:

Nethod 2:

Method 3:

Q/P1 elements

composite elements

18/Pl elements and filtering the pressures

by using only the centroida1 value

Exact Solution: lipIIL2(n)/lR = lOO J~~= 175.5942 ; h2 0.062S

IIp-p~ "L 2 (n) ~

IIp~ IIL2(n)/:R IIp-p~ IIL2 (n) /lR lipIIL2(Q)/~

Method 1 167,l254 20.0310 0.1141

Method 2 171. S448 36.3181 0.2068

Method 3 l71.S84S 26.6219 0.lS16

As a second example, we consider a Dirichlet Stokes Problem, which is

designed for numerical verification of the convergence theory for three

schemes considered:

a) Solution obtained with Q2/Pl elements.

b) Solution obtained with I8/Pl elements.

c) Solution obtained by averaging pressures of the 18/PI-scheme.

We consider the unit square domain partitioned in square subdomain, and

the following body forces ~ = (fl,f2) are applied:

212x + 24xy +

4 3l2x + 48x y

-2(x - xO) + a(x)

2 3 2 212y - 24x - 48x y - 72xy

2 2 3 4 2 3+ 72x y + 48xy - 24x y - 48x Y

2 3+ 48xy + 24y

+ 48x3y2

223 2f2 = 4x- 12x - 24xy - l2y + 8x + 72x y

4 322 3 4-l2y - 48xy - 72x y - 48x y + 24xy

where a(x) = -l if 0 ~ x ~ xO' a(x) = 1 if aO < x ~ l. Then

f(~,p) is defined by

2 3 2 2u = (-2x + 6x - 4x ) y (l - y)

2

and2

p = x - x - (x - x ) if o ~ x .::xoo 0

2 ifp = x - x - (x - x ) xo < x < Io 0

•

(~,p) satisfies:

-6u + p = f

ul = 0- r

div u o in n

in n

As before, we are interested by the plot of the pressure across a

section of the domain. Figure .6 shows the results obtained by partitioning

the domain n in 64 square subdomains. For this mesh, h is equal to

1/8, The computations are made with Q2-on 18-~lements. Whereas the Q2-

solution seems to be stable, clearly the 18-solution shows oscillations

around the exact solution. However, it is noted that both solutions

coincide at the centroid of the elements and this a~ain suggests that

the "smoothed 18-solution," obtained using only the pressure at the centroid,

2is stable, and may converge at a rate of O(h ).

Finally, Fig. 1,: confirms this suspicion showing the computed rate of

convergence is precisely 0(h2) for the pressure for the Q2-element, and

for the smoothed 18-element. However, it is also observed that the

Q2/Pl-pressures are considerably more accurate than the filtered 18/PI-

pressures for all mesh sizes considered.

-With the results from these examples we can conclude that

The Q2/Pl element is stable and the optimal L2-rate of convergence

2of the pressures of O(h) is attained.

The 18/Pl element yields unstable pressure approximations, but

these can apparently be stabilized considering only the values at

the centroids.

Spurious oscillations (checkerboarding) can also appear when

ker B* = IRh

Filtering the pressures in the 18/Pl-element by using only the

centroidal value leads to a pressure approximation which may

.9

converge in at a rate of 2O(h ); however, the accuracy of the

filtered scheme is quite inferior to that of the Q2/Pl-elements.

These computed results underline once again the critical role played

by the LBB-condition in studying the stability of finite element schemes

by reduced integration. These and other results we have computed also

indicate that the estimates obtained in Sectio~'\~ for the discrete

LBB-constant ~ are sharp. Indeed, the theoretical result that the use

of a composite element of the type employed here leads to a stable pressure

field, while not of great practical value, is fully confirmed by the

numerical results. This suggests again that these calculated estimates

of ~ are a good indication of the actual numerical performance of

these methods.

1.

2.

3.

4.

b,\

10

II

It

REFERENCES

BABUSKA, 1., "The Finite Element Nethod with Lagrange l1u1tipliers,"Num. Math., Vol. 20, 1973.

BERCOVIER, M., "Perturbation of Mixed Variational Problems - Applica-tions to Mixed Finite Element Methods, "R.A. I.R.O., Vol. l2,No.3, 1978.

BREZZl, R., "On the Existence, Uniqueness and Approximation ofSaddle-Point Problems Arising from Lagrangian Multipliers,"R.A.I.R.O., Numerique, Analyse, 8, 1974.

CAREY, G.F. and KRISHNAN, R., "Penalty Approximation of Stokes Flow,PART I: Stability Analysis, PART II: Error Estimates andNumerical Results", TICOM Report, 82-5, June 1982.

ENGLEHAN, M. and SANI, R., The Hathematics of Finite Elements, Editedby J.R, ~~iteman, Academic Press, Ltd., London, 1982,

ENGLEMAN, M., FIDAP User's Nanual, Boulder, Colorado, August, 1981.

FALK, R,S., "An Analysis of the Penalty Method and Extrapolation forthe Stationary Stokes Problem", Advances in Computer Methods forPartial Differential Equations, Edited by R. Vichnevetsky, AlCAPublication, pp. 66-69, 1975.

FORTIN, M., "An Analysis of Convergence of Mixed Finite Element Methods",R.A.I.R.O., Vol. II, No.4, 1977.

GlRAULT, V. and RAVIART, P.A., Lecture Notes in Mathematics 749, FiniteElement Approximation of the Navier-Stokes Eguations, Springer-Verlag, Berlin, 1979.

HUGHES, T.J.R., "Equivalence of Finite Elements for Nearly IncompressibleElasticity", J. Appl. Mech" March 1977.

LADYSZHENSKAYA, O.A., The Mathematical Theory of Viscous IncompressibleFlows, Gordon Breach, New York, 1969.

MALKUS, D.S., "Finite Element Analysis of Incompressible Solids",Dissertation, Department of Mathematics, Boston University, 1972.

MALKUS, D.S" "A Finite Element Displacement Model Valid for Any Valueof the Compressibility", International Journal of Solids andStructures, Vol. 12, pp. 731-738, 1976.

l~. NALKUS, D.S., "Finite Elements with Penalties for Incompressible Elasticity:A Progress Report", Illinois Inst, of Technology, Chicago, 1981.

IS' NALKUS, D.S. and HUGHES, T.J,R., "Mixed Finite Element }iethods - Reducedand Selective Integration Technique - A Unification of Concepts",Computer Methods in Applied Mechanics and Engineering, Vol. IS,pp. 63-81, 1978.

Ib aDEN, J,T. and WELLFORD, L.C.,Jr., "Analysis of Flow of Viscous Fluidsby the Finite Element Hethod", AIM Journal, Vol. 10, No. 12,pp. l59l-lS99, 1972.

17 ODEN, J.T., "R,I.P. Methods for Stokesian Flows", Finite Elements inFluids, Vol. IV, Edited by R. H. Gallagher et al, John Wiley andSons, Ltd" ~ondon, 1982 [presented at the Third Symposium ofFinite Elements in Flow Problems, Bnaff, Canada, June, 1980].

1\ aDEN, J,T., "Penalty Methods for Constrained Problems in NonlinearElasticity", Proceedings, IUTAM Symposium on Finite Elasticity,Edited by D.E. Carlson and R,T. Shield, (Lehigh 1980), MartenusNijhoff, The Hague, pp. 281-300, 1982.

'9. ODEN, J,T. and KIKUCHI, N" "Penalty Methods for Constrained Problemsin Elasticity", International Journal for Kumerical Methods inEngineering, Vol. 18, pp. 701-725, 1982;

20. aDEN, J,T .., KIKUCHI, N. and SONG, Y.J., "Penalty-Finite Elements Methodsfor the Analysis of Stokesian Flows", TICOH REPORT 80-2, Austin,1980: Also to appear in Computer Methods in Applied Mechanics andEngineering, 1982.

2l. REDDY, J.N., "On the Accuracy and Existence of Solutions to PrimitiveVariable Models of Viscous Incompressible Fluids", InternationalJournal of Engineering Science, 16-2l, 92l, 1978.

--,

18 I 18

Q2• 18

--Composite 1- 18

( a)

--

18 18 I 18 4

- --Q2

18I 18 I 8

- -- -18 18 l 18

-Composite 2 - 18

(b)

Figure t. Ex~mples of stable composite elements.

o

2

~...y

~

~f,

-"'--,..,

"'-1

~

I-

irx

Figure!, A mesh of 64 elements on an L-shaped domain.,

- - - -c .

- -0 0 c

- - -,

-- - - ""

0 0

- ... ... ... ... ...- - "" - - -c c 0 0 0 c 0 0 )

... I<- - - - .,. - ... ...- -c 0 ( > ~ ( c

... - - ... ... ... ...- - - - - -c 0 ) 0 c 0 ( 0

... ... - - ... -- - - -c 0 ) c 0

... - ... ... ... ... ... ... ... ...- - - - -

Figure 1. Mesh with composite-elements.•

Element

Element

P 44

42

40

38

34

32

\\\,\,\\

,\\\\,,

\,,\\\\,\\,\\\,

\,\\\,\\\\\\,\

,,, , ,

~' , ", ,

, I I, I ,

, , II I

CROSS SECTION A AIQ2

-----18

30o 0.5 1.0 1.5 2.0

XFigu/re 1,. Computed pressure profiles along Section M'.

42

40

38

36

34

32

CROSS SECTION A AlQ2 Element

----- Composite EL

30o 0.5 1.0 15 2.0X

Figure ·5. COr.lputed pressure profiles along Section AA'.

- Exact Solution--0- Q2 Solution---cr- 18 Solution

Figure ,6. Pressure profiles for second example.

,

Log II p_ph IIto Q2 Solutiono Smoothed 18 Solution

o

1/8 1/6 1/4Log h

Figure 1. Computed rate of convergence of the 18/PI - pressuresin L2 - with and without filtering.