of non-standard boundary conditions the solution of the. Stokes equations is unique. .... other higher-order versions of the Navier-Stokes equations and Maxwel]'s ..... by Hafez and Soil- man[26]. We construct the following quadratic functional: .... is equivalent to the following equations and boundary condition: in a,. (3.4a).
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,_