Theoretical Study of the Incompressible Navier-Stokes Equations by

NASA

Technical

Memorandum

106535

_/I,.?i

ICOMP-94-04

Theoretical Study of the Incompressible Navier-Stokes Equations by the Least-Squares Method

,.0 o o', N I '4"

N 0 0 U e"

Z

0 0 0

_D

Bo-nan

Jiang

Institute for Computational Lewis Research Center Cleveland,

Ching

Mechanics

Ohio

Y. Loh and Louis

A. Povinelli

National Aeronautics and Space Lewis Research Center Cleveland,

March

in Propulsion

Administration

Ohio

1994

\ =,,_.,.I= National Space

Aeronautics Administration

and

e..,.,/£, /

Theoretical

Study

of the Incompressible

Navier-Stokes

Least-Squares

Equations

by the

Method

Bo-nan Jiang Institute for Computational Mechanics in Propulsion Lewis Research Center Cleveland, Ohio 44135

Ching Y. Loh* and Louis A. Povinelli National Aeronautics and Space Administration Lewis Research Center Cleveland, Ohio 44135

Abstract

Usually Galerkin that

the

theoretical

method

which

the least-squares

differential

analysis

leads method

equations

since

of the

to difficult is a useful

it leads

Navier-Stokes

saddle-point

equations

problems.

alternative

is conducted

This

paper

tool for the theoretical

to minimization

problems

which

via

study

can

the

demonstrates

often

of partial be treated

by an elementary technique. The principal part of the Navier-Stokes equations in the first-order velocity-pressure-vorticity formulation consists of two div-curl systems, so the three-dimensional div-curl system is thoroughly studied at first. By introducing a dummy variable

and

is properly

by using

the least-squares

determined

and

elliptic,

method, and

has

a unique

employed to prove that the Stokes equations four boundary conditions on a fixed boundary This paper the solution of the

least-squares

differential method and

*National

also shows that of the Stokes

curl

equations

and

and

additional

by

operators

which

Research

Council-NASA

parts) play

Research

the

div-curl

is used

to prove

an essential

Associate

at Lewis

role

Research

that

The

the

same

div-curl technique

to derive

a high-order

conditions.

In this

Friedrichs'

inequalities

in the

Center.

system then

is

determined and elliptic, and that for three-dimensional problems.

of non-standard boundary This paper emphasizes the

method

boundary

shows

solution.

are properly are required

under four combinations equations is unique.

method

(integration

this paper

analysis.

paper

conditions application version

of

an elementary

related

to the

div

Introduction

The incompressible Navier-Stokes(NS) equations which govern the motion of viscous fluidsare among

the most important partialdifferential equations in mathematical

The Navier-Stokes equations are derived from the mass and momentum

physics.

conservation laws

and the linear local stress-strainrelation.While the physical model leading to the NavierStokes equations is simple, the situation is quite differentfrom the mathematical of view, and can be described by quoting the words from a mathematician: important

work

clone on these equations, our understanding of them

tally incomplete.'(sec Teman[1]). mathematical Even tions

our

understanding

remains

the

Here

boundary

review

jury

of the linearized

incomplete.

equations

recent

"Despite the

remains fundamen-

the major

difficultyin the

study of these equations is related to their nonlinearity.

of permissible Stokes

As pointed out by Teman,

point

we refer

have out

to the

conditions(BCs). been

the

on incompressible

is still

Navier-Stokes

subject flow,

regarding

the

fact that

The

there

boundary

and

has been

conditions

of constant

Gresho[2]

full story

equations

that

on mathematically

to the

Quoting

"To

equa-

no systematic applied

controversy.

states

the Stokes

the best

next

of our

permissible

study Navierfrom

a

knowledge,

BCs for the

NS

equations". Looking tions,

into

the classic

see for example,

find

that

standard

the

other

based

case

side,

study

in which

physicists

on physical

on the mathematical the

the

and

theory

Teman[1,4],

Ladyzhenskaya[3],

mathematicians

the

books

existence

velocity

engineers

and Girault

and uniqueness

components apply

of the Navier-Stokes

are

prescribed

the non-standard

Raviart[5],

and

of the

equa-

solutions on the

boundary

one

win

mainly

for

boundary.

On

conditions

often

intuition.

Usually the Galerkin method

is employed for numerical solutionand theoreticalanaly-

sisof the incompressible Navier-Stokes equations. The application of the Galerkin method leads to a saddle-point problem, and thus a difficult "Ladyzhenskaya-Babuska-Brezzi(LBB) condition" or "inf-sup condition" is involved[f-10]. In recent

years

the

developed

and

used

fully

LSFEM

based

on

vantages

over

other

resulting

discrete

solved

by matrix-free

to the

LBB

all dependent conditions

the

least-squares for the

first-order

condition, variables are imposed,

together are

iterative and and

element

method(LSFEM)

of incompressible

velocity-pressure-vorticity

methods: equations

finite

computation

symmetric

methods; thus test

with, and the

functions

may

be used; boundary

2

definite,

respect

only

simple

conditions

successThe

superior

ad-

linearization,

and

with

been

flows[11-19]. has

a Newton

of approximating

interpolation

numerical

formulation

example,

positive

choice

equal-order

no artificial

for

has

viscous

can

spaces

be

efficiently

is not

to a single algebraic for the

the subject grid

for

boundary

vorticity

need

be devised at boundaries at which the velocity is specified;accuratevorticity approxSmations are obtained; neither upwinding nor adjustable parameters are needed;all variables are solved in a fully-coupled manner, operator splitting turns out to be unnecessary. In addition,

we found

vorticity

formulation

a useful

tool

In

that

provides

previous

pressure-vorticity vorticity,

(three in most conditions are and than

of the solution

a saddle-point According

to the

equations

of the

Navier-Stokes principal

part

from

The consist since

Stokes

equations coupled

in electromagnetics.

systems.

For

works

counting

same

is just

the number

analyse

this

paper

paper

some

This

of unknowns we show

the

div-curl

we use

the

and

or not. that

system same

the

div-curl

by using

technique

sense)

the

to deal

(in the part

as that

of the

linear

nonlinear problem,

is the

verification

paper

the

Navierat least of the

on

system

solution

a similar

coupled

div-curl claim

of the

it is incorrect

whether

a system

Stokes

in the first and

In the

second

problem.

electric

by simply

determined

method.

in

of differential

of this problem,

is properly

that

are independent; equation

that

trivial,

out

electromagnetics equations

is tradi-

is not

to point of two

divergence-free

system

Stokes

formulation

div-curl

consist

two curl

least-squares

equations.

equations.

of the importance

with

of the

for the Stokes

its numerical

to judge

easier

ordinary

terms

principal

demonstrates

equations

Because

least-squares is much

Navier-

also

the

paper

existence

in the

books

only

electromagnetics

ignored.

the

which

It is worthwhile

engineering and

the

velocity-pressure-vorticity

equations

four

convective

three-dimensional and

flows

technique.

of this

equations.

of the

by an elementary

same

Stokes

system,

four

overdetermined,

is overdetermined

of this

and

The

Maxwell's

reason,

are

goal

for the

elliptic

on the first-order

by using

ellipticity

is the

is not

prove

problem

by the

in the first-order

and

on computational field

equations part

the

equations

or magnetic

main

systems.

unknowns

situation

some

the

written

The

velocity-

viscous

and

in [20], error analysis as that for the linear

conditions

div-curl

with

non-linear

equations

also

is not counted) boundary we give a further detailed

conditions

only

but

solenoidality

Based

conditions,

classification.

as an overdetermined

are three

Maxwell's

Thus,

boundary

elliptic.

dummy variable In this paper,

in [15], the

Navier-Stokes

is, the

incompressible

can be dealt

on the

problems

that

system

boundary

is determined

points.

considered there

often

solution

three-dimensional

to a minimization

according to the theory is essentially the same of the

of two

tionally

of the

singular

well-posedness

no effect

original

boundary

these

for numerical

Navier-Stokes

first-order

leads

and

equations

equations. Also Stokes equations away

under

explanation

have

the

three-dimensional

method

problem

Stokes The

the

for

velocity-pressure-

equations.

condition,

non-standard

least-squares

technique

that

compatibility

that

permissible

uniqueness The

showed

to make

on the first-order

Navier-Stokes

cases if the condition for the required for a fixed boundary.

of the

method.

the

we showed

a powerful

of the

we

based

for incompressible

and

be added

system

discussion

work[15],

sense,

should

elliptic

analysis

formulation

ordinary

method

not only

for theoretical

our

in the

the least-squares

elliptic, part

of

In fluid

dynamics

other

higher-order

often

useful

equations

versions

(see

the

merical

fields.

schemes.

divergence equation.

They

do its

in electromagnetic

tionM

boundary

squares differential solutions.

are the

method

on the

theorem:

equations),

and

most

appropriate

derived

equivalent

higher-order

if a vector

This

emphasizes

paper

boundary

condition

to a vector

that

paper

tegration

by

In Section

the

many

controversies

parts)

that

the

corresponds

components

with

three

In Section

to solving coupled

three

and curl

of addiequations the leastversions

method

then

of

of the

sys-

is based

and its

normal

vector

is zero.

this

operators

1, we use div-curl

and

that

independent

boundary

admits

(or spurious

an alternative

div-curl

together

with

components

higher-order

related

three-dimensional variable,

the

momentum

the

be zero)

version

of system.

boundary

conditions

for

an elementary

method

(in-

method.

inequality

a dummy

the

higher-order

in a domain,

permissibility

nu-

we show that

or the tangential

div-curl

alternative by applying

of the derived

is zero,

an equivalent

the

Friedrichs'

by introducing

system

over via this

as follows.

to prove

curl-free

component

and

without generating spurious with the least-squares weak This

boundary

the div

to derive

can be resolved

is organized

apply

normal

equation

2, we show

elliptic

one must

(either

equations

The

that

and

flow

selection

We also show

systems.

on the

fluid

to the

careful

way to derive conditions associated

are

"primitive"

solutions

the solution

equations.

is divergence-free components)

differentia]

We believe

boundary equations

equations

differentiation

spurious

as a by-product,

consistent

why

formulation;

operator

With

reasons

the

for devising

by simple

words,

In this paper,

from

is obtained

cur]

be generated.

to derive

(or tangential

div-curl

additional

Maxwel]'s

velocity-pressure

the

obtained

other

may

a systematic

component

ity

(and

provides

tematic

and

in

good

regarding

point

equation

in the

equation

and

all derived insights

starting

by applying

a derived

equations.

are

Poisson

equation

equations and corresponding That is, the Euler-Lagrange

formulations

derived

as the

waveguides)

the original

method

serve

are some

equations

additional

the pressure

progenitors,

conditions

will also solve

also

there

These

offer

is obtained

is that

than

often

momentum

equation

A key issue

solutions

Navier-Stokes

by Gresho[2]). and

to the

transport

modes

of the

For example,

operator

vorticity

as in electromagnetics

discussion

by differentiation

electromagnetic

more

as well

the

to the

div

system

curl

is properly

least-squares

Poisson

conditions.

and

determined

method

equations

In Section

operators. for the

of three

veloc-

2.4, we introduce

the

div-curl method to change the low-order partial differential equations into an equivalent higher-order form. In Section 3, we study the div-curl system with a different boundary condition. about

In Section

the

inequality.

orthogonal

4, we use

In Section

5, we show

velocity-pressure-vorticity straint we show problems

condition that and

list

results

divergence

number

that of the

the

vorticity

combinations

in Section

and

boundary

problem

determined should

the

theorem

Friedrich's

second

written and

be zero)

conditions

of boundary 4

1-3 to prove

use it to establish

Navier-Stokes

is properly

of permissible

all possible

obtained

of vectors,

formulation

(the

the

the

decomposition

in the first-order

elliptic

when

is supplied.

is four conditions

for three for

the

the

con-

In Section

6,

dimensional Stokes

prob-

lem. In Section7 we prove that Stokes

problem

has

pressure-vorticity problem order are

via

a unique

derive

scalar

functions.

10.

some

method. by

1.1

using

the

the

boundary div-curl and

the

give

two

velocity-

on

remarks

vector

Poincare

Stokes second-

Concluding

equalities

the

for the

conventional

method.

useful

conditions

second-order

conditions

9, we derive

all

formulae,

Basic

boundary

8, we derive

corresponding

we list

Green's

1.

nonstandard

In Section

In Appendix

important

different

In Section

the

formulation

in Section

briefly

and

least-squares

velodty-vorticity given

four

solution.

formulation the

under

operations,

inequalities

for

Inequalities

Notations Let f_ C _3 be an open

f_, _ be a unit

outward

normal

and

_2 be two orthogonal

and

maintain

and

the

necessary. with

the

inner

vector

we often

I" is smooth

L2(f_)

domain

tangential

simplicity

boundary

bounded

denotes

the

with

on the

boundary,

vectors further

enough, space

a boundary

_ be a tangential

to F. In order assume

although

P, x = (x,y,

that

to pay

the

in many

of square-integrable

cases

vector

attention

domain

z) be a point

to F, and _-_ to basic

fl is simply these

functions

restrictions

defined

in

ideas

connected are

not

on ft equipped

product

(_,,v)= / _,vd_

_,v

E Z2(n)

Jn and

the

norm

I1=11_,. - (=,=) H_(ft)

denotes

the

Sobolev

space

of functions

up to _. I" I_,_and ll"li_,_denote For vector-valued

functions

_ with

= _ L2(fZ). with

square-integrable

the usual seminorm and m components, we have

the

inner

product

u,v _ L2(_)"

(_'_) = fn _" vdft and

the

corresponding

norm w/g

--

= _ II=JII_,n. I1_11_,_

2

2 I1_110,. = _ ll_ll0,o,

j=l

j=l

5

of order

norm for H_(ft), respectively. the product spaces

L2(n)m, H_(_)'_ with

derivatives

Further

we define

< u,v

and

the

corresponding

e H_/2(r), _ e H-_/2(r)

>r = fruvds

norm

II_ll_/,,r=< ",_ >r. When

there

is no

inner

product

and

Througout different

values

Lemma

1.1.

Every

chance norm

the

Let

confusion,

C denotes

_ of H 1 (ft) 3 with

Proof.

and

Using

a positive

ft be a bounded

open

R2 denote Green's

the

measure

ft or I' from

the

constant

dependent

on

_ with

possibly

subset

of ]t 3 with

a piecewise

C 1 boundary

r.

_ x _ = 0 on r satisfies 1

I_1_+ fr( _ R1

omit

appearance.

1

where

we will often

designation.

paper

in each

function

for

+

the

principal

formulae

(B.2),

)_2d8--IIV radii (B.3)

+ IIV x

of curvature and

(B.7),

_[102,

(1.1)

for I'. and

Equality

(A.4),

we have

llv. _11]+ tlV× _11_ = (v. _, v. _) + (v x _, v x _) =(_, -v(v._))+

< v._,

n._ > +(_, v(v._)-z_)+

< v x _, _x_> 0-_

=(V_,

V_)++-"

(1.2)

Obviously

(v_, v_)= I_1_. Now

we turn

to the

boundary

integral

terms,

n x u = 0 on I' implies

_=U_ where using

U is a scalar function which (A.1), the boundary integral

that

onI',

depends on the location on the boundary terms in (1.2) can be written as follows:

f{vv.

(v_) - u_OV }d8 6

surface.

By

/,

= _r U_V It can

be verified

that

on a curved

"_tds.

surface 1

v._= where

7 is the

mean

1

n_+ N =2%

curvature. 13

Lemma Every

1.2.

Let

function

_q be

a bounded

_ of H 1 (f_)3

with

open

R1 < R < R2,

Proof.

In this

case

in which

we still

R1 and

have

of the

triple

scalar

a piecewise

C 1 boundary

(1.2).

= IiV._ll0 _ + IIV× _lf0 _, R2

denote

Since

n.

= U_

By virtue

of Ns with

the

principal

= = 0, we may

radii assume

(123) of curvature that

on F.

product

(v × _).(_ x _) =_.(_ × (v × _)) and

using

(A.5),

the

boundary

integral

fr

=

Jr

F.

ft. u = 0 on 1" satisfies

I_1_+ f_ _ds where

subset

terms

in (1.2)

{_.(_xVx_)

can be written

2On 1 O-_2 }a,

{_.(v(_ _)- (_v)_) 2 1 o_-_ o,, _ }ds = ./r{_. (-(_v)_)}a_ = ./r{u_" (-(_v)_)}a_ 7

as

for r.

So the Lemma direction of _.

is proved.

Here

R is the

radius

of curvature

of the boundary

surface

in the U

Since derive

the

curvature

immediately

1/R

the

is always

following

positive

measure.

Every

2.

Suppose

function

that

_ of H 1(f_)3

the

simply

Theorem 1. then _ = 0.

If _ E H 1 (ft) 3 satisfies

Proof.

Theorem

From

boundary

surface

convex open subset of _s, and (Pl, F. Either £1 or F2 may be empty

lul_ ---II v'ull0 Theorem

the

is convex,

we

theorem:

Theorem 1. Let _ be a bounded and of the piecewise C 1 boundary surface positive satisfies:

when

with n"

_ + IIv

connected

V.

P2) be the pieces or with strictly

u = 0 on F1 and _ × _ = 0 on

× ull__ and

F2

(1.4) r satisfy

_ = 0, V × _ = 0, _-

the

lrl

same

= 0

assumption

and

_ ×

lr,

in

-- 0,

1, we have

-o, 14

(3.4c)

_ on I', or _ is perpendicular on

variable

and boundary

on F. with

By intro-

as

in l'l,

any boundary

is, _ is parallel

only

(3.3a)

Vx(V¢+Vx )=o

Since _ x _ = 0 on F, that necessarily have

has

algebraic

are correct.

onF.

is equivalent

problems

to F, we (3.5)

say, _.

(V x _) > 0 at a point

in which

s is a small

positive

constant.

From

0=_.dl=_ where

c is the boundary Taking

into

contour

account

the

Stokes

theorem

we have

a contradiction:

(Vx_)._ds>0,r

of OF.

(3.5),

(3.4a)

and

(3.4c)lead

/k_b=0

in

to it,

(3.6)

0¢ = 0 o= r.

(3.7)

On That and

is, ¢ is a constant, four

unknowns

or V¢

is equivalent

determined,

so is System

(3.1).

components

of _ must

be zero

the

variable

dummy

conditions

which

are

4.

Every

consistent

with

(3.1).

boundary

on the be

System

System

condition

surface

F.

specified,

In this

with

case

four

is elliptic

requires

that

eUiptic

are

only

equations

and

properly

two tangential

no boundary

there

Decomposition

_ E H 1(fl)3

(3.3)

(3.3)

(3.1c)

so altogether

a 2 x 2 first-order

Orthogonal

vector

Therefore

to System The

¢ should

4. Theorem

= 0 in ft.

condition two

on

boundary

system.

of Vectors

has the orthogonal

decomposition:

= Vq + V x 5, where q • Proof. find

H2(n)/

we note q and

virtue

5-

and 7 • H (n)

that This

of Theorem

(4.1)

for

the

theorem 2, Equation

purpose shall (4.1)

of investigation

be proved

in this

by directly

is equivalent

paper,

aeeking

to the following

for

there Vq

equations

is no and and

need_ to

V × ¢.

By

boundary

condition: V-(Vq

+ V × _ - _) = 0 in l'l,

(4.2a)

V × (Vq+V×¢-_)=0ina, _.(Vq+Vx¢-_)=O Taking

into account

V. V x ¢ = 0 and

(4.2b) on

V x Vq = 0, System

Aq = V. Vx(VxS)=Vx_ina,

_ in f/,

P. (4.2)

(4.2c) can be written

as follows: (4.3a) (4.3b)

15

_.(vq+ We may

solve

the following

Neuman

v × ¢)=.._ problem Aq

of Poisson

= V.

i.e.,

the

boundary

an arbitrary Now

condition constant

¢ should

(4.5)

least-squares

by the

decomposition (8.4)

to obtain

q:

onF,

(4.4b) supplied.

Although

q, Vq is uniquely

q is not

unique,

determined.

satisfy

determined

Using

into

equation

(4.4a)

is additionally

can be added

V x ¢ in System of the

(4.4b)

(4.3c)

_ in _,

_.Vq=n.u where

on r.

and

may

v .(v × ¢) = 0 i, a,

(4.5a)

v × (v × ¢)= v × _ i_ a,

(4.50

_. (v × ¢)= 0 on r.

(4.5c)

be considered method

(4.1)

is proved.

(4.5c)

we have

as an described

unknown

vector

in Section

and

can

2. Therefore

be the

uniquely validation

(vq, v × _) =< _. (v × _),q >= 0, m

that

is, Vq

and

V × ¢ are

orthogonal. D

Since q is the following regularity

solution of the Neuman result [27,28]:

problem

of Poisson

equation

(4.4),

Iq12___ C{l/V._ll0 + I1_-_lt,/2,r}. Since

V × ¢ is the

least-squares

solution

of (4.5),

by virtue

of (2.7)

Iv × ¢1, ___ I}V× _110. Hence

by using

(4.1),

(4.6)

and

(4.7)

we obtain

II_ll,- II_ll_.

(7.22)

is indeed

to prove

v) ___I1_11_.

together

CB(V,V) This

the

(7.21)

yields

ca(v, Combining

(7.20)

to

ca(v, Since

_ IIVx _1120.

we finally

obtain

that

> I1_11_ + Ilqll_+ I1_11_.

bounded

that

(7.23)

this

below

problem

(7.24)

in H 1 norm has

a unique

and

thus

coersive.

solution

that

the

developed

Con-

satisfies

the

bound:

I1_11, + IIr,ll_+ I1_11_ I1_11_---I1_11o_-

(7.35)

(7.32) with (7.35) we have

CB(V, v) _>I[_ll_

(7.36)

and

CB(V,V)

> 11711o 2

or

C(B(V,V))_ 28

_ I1_11o.

(7.37)

From

(7.33)

we have

(7.38)

(n(v, V)) _ >__ II- _ + V x _110. Combining

(7.37)

with

(7.38)

and

using

the

triangle

inequality

lead

to

C(B(V,V))_ _ I1_1t0 + II-_ + V x _110>--IIVx _110, that

is

(7.39)

C(B(V,V)) > IlVx _11 _ __

Combining

(7.34)

with

(7.39)

leads

0

°

to

(7.40)

CB(V,V) > IIVx _11_+ IIV"_110 _Since

n • v = 0 on F, from

Theorem

3 we have

the inequality

(7.41)

C(llV x _110 _+ IlV, _11_)>-II_ll_. Combining

(7.40)with

(7.41)yields

CB(V, V) _ I1_11_. Combining

(7.31),

(7.36)

and

(7.42)

together

we finally

(7.42)

obtain

(7.43)

CB(V, V) >__ ll_ll_+ Ilqll_+ I1_11_Therefore, '/.3

the

p=0,

u_l

This

boundary

boundary implies

=0,

u_2=0,

are

cases

in the

so it is all right By virtue boundary

w,_=0(p=0,_×_=0, may

be used

prescribed.

we have

¢ in advance, that

is proved.

As

specified

so only

present to specify

of Theorem

case

three

the

for the

four

that

onF

well developed are

too many

condition

conditions condition

exit

boundary.

Here

four

in §3.3 _ x _ = 0 on F analytically

there

boundary

boundary

no boundary

n.o:=0)

mentioned

_- (V × _) = 0 on F. It seems

previous

variable

of B(V,V)

condition

conditions that

In the show

coersivity

boundary

are needed.

is needed

conditions.

¢ = 0 on F for the for the

In the

dummy

following

dummy

we

variable

¢,

conditions.

2, Equation

(5.3c)

is equivalent

to the

following

equations

and

condition: (7.44a)

Vx(-_+V¢+Vx_)=Oinl2,

V.(-_+V¢+V

x _) = 0 in _,

(7.44b)

n.(-_+V¢+V

x_)

(7.44c)

29

= 0 on r.

Taking

into

account

(5.3b)

and

V.

V x _ = 0, Equation A¢=0

Taking

into

account

in

w,_ = 0 on F and (3.5),

(7.44b)

becomes

the

fl.

(7.45a)

boundary

condition

(7.44c)

becomes

0¢ = 0 on r. Equation

(7.45a)

and

conditions

in the

eliminated

in Equation

These i.e., the let's

present

boundary

tangential

consider,

(7.45b)

imply

case

that

(7.45b)

¢ is a constant

automatically

or V¢

guarantee

that

-- 0 in fL

the

dummy

Therefore,

variable

four

¢ can

be

(5.3c). conditions

velocity

for example,

correspond

components the

to those

and

surface

the

with

in the

normal

_ = (1,0,

velocity-pressure

stress 0).

are prescribed.

Since

formulation, To show

this

v = w = 0, we have

_V

0y

= 0,

_W m

_

O.

Oz Hence

from

the

continuity

of velocity

we know

--

_

that

0°

Oz Therefore, _u

a_ = p + 2vw--

= 0,

Ox

that

the

is, the

normal

stress

The least-squares coersivity of B(V,

B(V,V) Since

is zero.

method minimizes V). We have

the

same

functional

(7.2).

We shall

- IlVq+ V x _11o_ + IIV._lto_+ I1_- V x _11_+ IIV._II_.

q = 0 on 1", by virtue

of (7.26)

(7.46)

we have

IlVq + v x _llo _ = IlVqll_+ IIV x _ll_Thus

now prove

(7.47)

we obtain B(V,V)

---llVqllo _ + IIv x _llo_ + IIv ._llo _ + II_-

3O

v

× _llo_ + IIv ._llo_.

(7.48)

Therefore

S(V,V) Since

q = 0 on P, from

Poincare

_ IlVqll_-Iql_.

inequality

(C.2)

(7.49)

we have

(7.50) Combining

(7.49)

with

(7.50)

yields

CB(V,V) From

(7.48)

we know

that

B(V,V) Since

(7.51)

_ [Iqll_-

n • _- = 0 on F, from

(7.52)

_ IIv x _11o _ +'llV-_llg-

Theorem

3 we have

the

inequality

(7.53) Combining (7.52) with (7.53) yields CB(V,V) From

(7.48)

we also

know

_>

(7.54)

I1_11_.

that

(7.55)

From

From

(7.54)

(7.55)

B(V, V) _ IIV. _11o 2.

(7.56)

CB(V,V)_ _ I1_11o.

(7.57)

we have

we know

(7.58)

B(V, V)_ >__ II- _ + V x _ll0Combining

(7.57)

with

(7.58)

CB(V,V)] that

and

using

the

triangle

inequality

lead

to

I1_110 + II-_ + v × _110>--IIV× _110,

>

is

CB(V,V) > ItV× _11_. Combining

(7.59)

with

(7.56)

(7.59)

yields

CB(V,V)

> IIV× _11_+ IIV-_ll_. 31

(7.60)

Since_ x _ = 0 on 1",from Theorem 3 we have the inequality

Combining

(7.60)

and

(7.61)

C(llV

x _11o _ + IIv "_11_) > I1_11, _.

leads

to

(7.61)

OB(V, v) > I1_1t, _. Combining

(7.51),

(7.54)

and

(7.62)

together

we finally

(7.62) obtain

the

coersivity

CB(V,V) ___ Ilqll_+ I1_11_ + I1_11_If we really boundary case,

the

specify

conditions

the

are

least-squares

dummy

needed,

method

S(U)=llVp+Vx_-Yllo

variable

and

that

minimizes

_+llv.zllo

¢ be zero

on

F in advance,

w,, = 0 can be imposed the

following

(7.63)

then

in a weak

only

sense.

three In this

functional:

_+11 _-vx_llo

_+11 v._llo

_+ll_._ll_/2x.

(7.64)

We have

B(u, v) = (vp + v x _, vq + v x _) + (v. _, v. _) +(_- v x _,_- v x _) + (v._,v._)+ L(V)

< _ .n,_.n >,

= (f, Vq + V x e),

(7.65) (7.66)

and

B(V,V) Since

.....IiVq+

V x _11_, + IIV _11_ + I1_ V x _11o _+ IIV _t1_, + I1_ -_ll,/_,r-2(7.67)

q = 0 on F, by virtue

of (7.26)

we have

IlVq + V x _ll0_ = IlVqll0_ + IIv Thus

x _11_-

(7.68)

we obtain

B(V,V)

= IiVqllg+llV

x_llo _ +11 v._llo

_ +1t _-

v

x_llg+llV._ll

g +11 _.rll,/2,r.2

(7.69)

Therefore B(V,V) Since

q = 0 on F, from

the

Poincare

___IlVqll0 _ = Iql_-

inequality

(C.2)

Clql_ >_ Ilqll,. 32

(7.70)

we have

(7.71)

Combining

(7.70)

with

(7.71)

yields CB(V,

From

(7.69)

we also

know

us expand

the

(7.72)

_> ]lqil_-

that

s(v,v)

Now let

V)

(7.73)

> Ilv × _11_+ iLv'_lLo _+ li_" Itl/_,r,

fourth

B(V, V) > il_- V × _tl:o,

(7.74)

B(V, V) > IIV"_il_0-

(7.75)

term

in (7.69).

Since

n x v = 0, using

Green's

formula

(B.5)

we have

I1_- v × _li_= li_lL0 _+ ilv × _lf_- 2(_,v × _) = IL_li_0 + ]iv × _ll_- 2(v × _,_). (7.76) Therefore

(7.69)

becomes

B(V, V) = liVqII_+ IIV× _llo 2+ IIV-_llo _+ II_ll_+ IIV× _11_ -2(V Since

_ x _ = 0 on F, from

(7.77)

_,_) + IIV•_lIo _ + I1_•-rl11/2,r.2

Theorem

C(ll v Multiplying

x

by C _ and

3 we have

the

× _11_o+ IIv "_li_) taking

into

account

(7.77)

inequality:

> I1_11_-> II_llo_" (7.78)

(7.78)

we have

C_B(V,V) >_C_ll_ll_o + C_(IIV× _11o _ + IIV._11o _)-2c_(v

× _,_)

or

CB(V,V) > C_lt_ll_+ CIl_llg 2C_(V × _,_), From

(7.73)

we have

c_B(v,v) Adding

(7.79)

and

(7.80)

_>c_llv × _11_.

(7.81)

obviously

(7.80)

yields

(c _+ C_)B(V,V) > C_l[_l[0 _+ CIICV × _ - _110 _. From

(7.79)

(7.81)

we have CB(V,V)

>

33

It_11o _.

(7.82)

Combining (7.73)

with

(7.82)leads

to

CB(V, v) _ IIVx _ll0 _+ IIv•_ll0 _+ II_l[0 _+ II_•-_lll/2,rFrom

Theorem

5 we know

that

(7.83)

_ satisfies

(7.84)

C{ll_ll0+ IIV._11o + IIV x _110 + II_'_ltl/=,r} > tl_ll,. Combining(7.83)with (7.84)leadsto CB(V,V) > I1_11_. From

(7.82)

we have CB(V,V)]

From

(7.74)

(7.85) (7.86)

> I1_110.

we know

CB(V, V)-_> II- _ + V x _110. Combining

(7.86) with

(7.87) and using the triangle

(7.87)

inequality

lead to

CB(V,V)-_ ___ I1_110 + I!-_ + V x _110_>IIVx _110, that

is

CB(V,V) _>IIVx _11o _. Combining

(7.75)

with

(7.88)

and

considering

(7.78)

(7.88)

yield

CB(V, v) >_11_11_. Combining

(7.72),

(7.85)

and

(7.89)

together

we finally

(7.89)

obtain

the

coersivity

CB(V, v) _>Ilqll, _+ II_ll_+ I1_11_. 7.,1

u,l=0, For

boundary the steps

7.5

u,2=0,

the

same

permissibility

this

w,_=0(_x_=0,

as explained

of ¢ is specified.

u_l=0,

Obviously the

reason

condition in §7.1.

u,_=0,

w,l=0,

in §7.3, The

u_2=0(_._=0, is a standard

cannot

nxw=0) in this

coersivity

case

(7.90) onF

¢ -

of B can

0 is guaranteed

be proved

by just

even

no

following

_x_=_)onF permissible

be proved

by the

boundary

condition.

elementary

method

34

However, presented

it seems in this

that paper.

Fortunately,

one

may

rely

on

the

ADN

theory

to fulfil

the

task,

see Bochev

and

Gun-

zburger[31]. 7.6

p=0,

w,_=0,

Using p and

_.

boundary this

the

w_l=0,

boundary

However,

the order

equations

problems

which

p = 0,_,_

and

(8.3f)

be used

(see the

the

with

can be rewritten

(Vp+V

onl" solve

Equation

by solving

discussion

(8.3f)

in the next

(6.1) with

to obtain

the

section).

natural

Therefore,

of boundary.

Method

how

can

determined

Equations

Least-Squares

associated

= 0 one

on a part

Euler-Lagrange

to understand

Lagrange

n.w=0,_×_=0)

be uniquely

in (8.3g) can only

8.

In

conditions

_ cannot

conditions

combination

w_2=0(p=0,

Associated

for

the

least-squares

Stokes

method

the least-squares

with

weak

Equations works,

we

formulation

derive (7.3)

the

Euler-

for the

Stokes

as

x_-],

Vq+V

(_-Vxu,

x_)+(v._,V._)+

r-Vx_)+(V._,V._)=0,

or

(Vp+Vx_-f,

+(vp+v

Vq)

x _-?,

+(v._, +(_-

v x_)

v._)

v x _, _)

-(_-Vx_,Vx_)

Using

Green's

formulae

(B.1),

(B.3)

+(v._,

v._) =0.

and

from

-(V.(Vp+Vx_-?),

(B.5)

(8.1)

(8.1) we have

that

q)+

+(v x (vp + v x _ - ?), _)- < _ × (vp + v x _ - ?), _ > -(v(v._),

_)+ < v ._, _._ >

+(_ - v x _, _)

-(v x (_ - v x _), _)+ < _ x (_- v × _), _ > 35

-(v(v._), From

(8.2)

boundary

after

simplification

we

_)+ < v._, _._ >=0. obtain

the

following

(8.2)

Euler-Lagrange

equations

and

conditions: Ap = V.

f in £,

(8.3a)

q = 0 or _. (vp + v x _ - y) = 0 on r, A_--_+V

_x_=O

x _ = -V

(S.3b) (8.3e)

X f in _,

(8.3d)

or _x(Vp+Vx_-7)=_onr, n.r=0

or

(8.3e)

V._=0onF,

A_ + V x _ = _ in £,

_x_=_

o_ _x(_-Vx_)=_o_r,

n.v=O Equation second-order ary serve

(1)

as the

n-u,

reveals

elliptic

conditions

boundary

(8.3) serve natural

conditions

that

equations as the

the

and

boundary

or

V._=Oon

least-squares seven

essential

for different

(8.3/)

conditions.

I',

weak

boundary

boundary

(8.3g) (8.3h)

formulation

conditions conditions

In the

in which and

following

cases:

n x _ given

,. (vp + v x _-?)

= o,

V._=0,

_x(_-Vx_)=& (2)

p, _-_,

_. _ given

x (vp+ v x _-?)

= _,

_x(_-Vx_)=& (3) p, _ x _, _. _ given x (vv+v

x _ - .f) = _,

V._=0. (4)

_xu,

nx_given

_.(vp+

v x_-l) 36

=0,

some we list

corresponds the original first-order the

to seven boundequations

combinations

of

(5) u _ven

vp+ v x_-7

=_,

V._=0. (6)

p, n-w,

n ×_given

_x(_-v

x_)=_,

V._=0.

We emphasize

again

pressure-vorticity

that

the least-squares

formulation

(5.5)

does

Only if someone would like to use, second-order velocity-pressure-vorticity additional

natural

boundary

conditions

that

in the

second-order

of the

pressure

We notice the

solution

velocity of the

and

the

present

continuity including boundary operator

standard

Poisson

vorticity

through

second-order

of velocity these

two

condition

and

The

us first boundary

the

but

also

not

need

on the

additional

first-order

boundary

velocityconditions.

finite difference method to solve (8.3a), (8.3c) and (8.3f), should

the the

be included. velocity-pressure-vorticity equation

the

(8.3a)

boundary are

solenoidality

that

formulation

is coupled

conditions.

with The

(1) it guarantees

constraint

equations;

for

based

any

example, the formulation

formulation

divergence-free

is self-adjoint

9. Let

for

method

on the

boundary

the not

solution

significant without

only

conditions;

for the

(3)

of the

advantages

satisfaction

vorticity

(2) it is suitable

non-standard

the

in general

of the explicitly standard

the

differential

problem

with

(symmetrical).

Div-Curl

consider

the

Method

for

following

the

first-order

Navier-Stokes system

of the

Equations Stokes

the

condition: Vp+V

x-_

= f in _,

V .-_ = O in _,

(9.1a) (9.1b) (9.1c)

-_+Vx_=0in_, V . _ = O in _, u=ur

ont.

37

(9.1d) (9.1_)

Of course

the boundary

data

_r

should

satisfy

the global

rn From

Theorem

2 we know

that

System

mass

conservation:

(9.1f)

= 0.

(9.1)

is equivalent

to the following

system:

Vx(Vp+Vx_-f)=0infl,

(9.2a)

v. (vp + v × _ - f) = 0 in a,

(9.2b)

_.

(9.2c)

(Vp

+ V x _-

f)

= 0 on F,

v x (_- v x _) = 0 in [2,

(9.2d)

v. (_ - v x _) = 0 in n,

(9.2e)

x (_-

V x _) = 0 on £,

(9.2f)

V . _ = 0 in [2,

(9.29)

v. _ = 0 in [2,

(9.2h)

= u-r on F. Taking (9.2)

into can

account

(9.2i)

V x Vp = 0, V. V x _ = 0, V. V x _ = 0 and

be simplified

Equality

(A.4),

System

as:

A_ = - V x -] in [2, Ap

x _-

n x (_-V

Since

in §2.3,

Equation

(9.3a)

Equation conditions

implies

(9.3e)

O in [2,

(9.3f)

V . _ = O in [2,

(9.39)

= ur

boundary

(9.3d)

x_) =0 on r,

V-_=

the

y) = o on r,

+ V x -_ = 0 in [2,

A_

As explained

(9.3b) (9.3c)

= V " f in [2,

_.(Vp+V

(9.3f) and velocity.

(9.3a)

(9.3g) (9.3e)

on F,

(9.3h)

can be eliminated, and

(9.3h)

guarantee

since the

Equation divergence-free

(9.3d)

and of the

that

A(V. _) = 0 i,_ [2, 38

(9.4a)

if we specify

that

v. _ = 0 on r, then

V._

-

0 in f_, that

we can

replace

replace

(9.3e)

is, the solenoidality

Equation

(9.3f)

by

the

(9.4b)

of the vorticity boundary

vector

condition

is guaranteed.

(9.4b).

Therefore

Furthermore,

if we

by that - V x _ -- 0 on r,

then

(9.3e)

and

(9.4b)

all are

velocity-pressure-vorticity

satisfied.

formulation

Finally for the

(9.4e)

we obtain Stokes

the

conventional

second-order

problems:

(9.5a)

x -] in £,

A-_ = -V

(9.5b) (9.5c)

_-Vx_=0on£, _=u-r

(9.5d)

onI',

(9.5e)

Ap = V . f in £,

v x _-?)

_. (vv+ From from

that

(9.5)

we understand

of the

called

the

a part

of boundary For

following

pressure,

the

reason

formulation. pressure

Navier-Stokes

non-linear

the calculation

for this

velocity-vorticity the

that

of the velocity

in the

literature

We note

that

and

vorticity

Equation this

(9.5a)

decoupling

is decoupled and

does

(9.5b)

not

are

hold

if on

is prescribed.

equations,

momentum

(9.5/)

= 0 on r.

Equation

(9.1a)

in system

(9.1)

is replaced

by

the

equation[2,15]: 1 x _ + Vp+

where the

p should

Stokes

be understood

equations

we obtain

as the the

total

_ee V x _, = f pressure.

second-order

Following

(9.6) the

velocity-vorticity

Re/x_ - V x (_ x _) = -V

x ] in £,

A_ + V x "_ = -0 in £t, _-Vx_=Oon£, -- ur on F,

39

in £, same

steps

as those

for

formulation: (9.7a)

(9.7b) (9.7c) (9.7d)

hp+V

x

iv

_.(_x_+VP+Re

10.

The

least-squares

a powerful

technique

of the div-curl the Navier-Stokes not

Since

systems,

four

given, the problems,

for numerical

problems

by using

element

equations equations

on

the

first-order

solution,

part

conditions

(9.7f)

on F.

Conclusions

three-dimensional

principal

conditions

but

of the

differential

also

a useful

equations

Navier-Stokes

on a fixed

equations

tool

for

should

equations

boundary

are

methods generating obtained and are

with

The

method.

equal-order

least-squares

to obtain spurious by the suitable

Consequently, and

(if three

for the

version

method of the

the self-adjoint

study and are

boundary

of two

div-curl

conditions

are

for three-dimensional combinations of non-

an optimal

div-curl

least-squares method automatically for any boundary conditions.

4O

two

Navier-Stokes

the corresponding has

the

derived

Specially,

have

to be permissible

interpolations method

a high-order solutions.

proved

only

equations formulation

consists

needed

is not

theoretical

equations. The div-curl velocity-pressure-vorticity

div-curl

are rigorously

the least-squares

method

all unknowns.

without

the

=0

dummy _ = 0 on F should be counted as the fourth one) and two for two-dimensional problems. Four different boundary

consistent

The

boundary

standard finite

based

(9.7e)

x_-f)

equations and the Navier-Stokes equations in the first-order

overdetermined.

conditions.

for

method

= V ..f

rate are

of convergence systematic

differential

second-order satisfy

least-squares

the

and

equations differential

divergence-free

Appendix

A. Operations

on Vectors

v.(@)

(A.1)

= qv._+Vq._,

v x (@) = qV x e+Vq

(A.2)

x _,

v. (_ × _) = (v × _). _ - _. (v × _),

(A.3)

V x V x _ = V(V. _) -/x_,

(A.4)

× (v × r) = _v(_ _)- (_v)_.

(A.5)

_5

B. Green's Assume divergence

Formula that

u, v and

theorem

lead

q are

_ = Vp

into

(B.1) (Ap,

Substituting

q = V.

enough.

q)+(_,

Vq)=.

_ = Vq into (Vx_,

using

the

Gauss

(B.1)

the

Vq)

=
.

(B.2)

v(v._))=.

(B.5) Vq)=-

(B.3)

to Vq)=.

(B.4) lead

to

(v × _, _)- (_, v × _)=< n × u, v >. Substituting

and

yields

leads

(V×_, Integrating

(A.1)

yields

(v._, v._)+(_, Replacing

Integrating

to (V._,

Substituting

smooth

(B.5)

yields =.

(B.6)

Replacing _ by V × _

in (B.5)

yields

(B.7)

(Vx ,

C. Poincare Let

Inequality

Ft be a bounded

domain

with

IIPlI_ -< C{llVPllo

a piecewise

_ + (f,

IIPlI,_