16

Existence of
-Weak Solutions by G2

One may be tempted to believe that physical laminar flows correspond
to smooth mathematical solutions (of the Navier–Stokes equations) and
turbulent flows to non-smooth solutions (Oseen in Hydrodynamik ...).

On peut v´

erifier en outre que l’´

energie cin´

etique totale du liquide reste

born´

ee; mais il ne semble pas possible de d´

eduire de ce fait que le mouvement

lui-mˆ

eme reste r´

egulier; j’ai mˆ

eme indiqu´

e une raison qui me fait croire `

a

l’existence de mouvements devenant irr´

eguliers au bout d’un temps fini; je

n’ai malheuresement pas r´

eussi `

a forger un exemple d’une telle singularit´

e

(Leray 1934).

We will now discuss in a little more detail the Struggle for Existence.(Darwin)

16.1 Introduction

We now show that we may construct
-weak solutions of the NS equations
using stabilized Galerkin finite element methods in the form of G2. We do
this in order to highlight a basic property of a G2 solution, which is designed
so as to have a a small residual in a weak sense, and thus may pass as an

-weak solution for a certain
depending on the mesh size. We do not here
give full details of the formulation of G2, e.g. concerning the use of continuous
or discontinuous Galerkin for the time stepping, but focus on the basic role of
the stabilization in G2, and give a complete description of G2 in Chapter 28.

In the discussion of the Clay Prize Problem we commented that an alterna-

tive way of proving existence of
-weak solutions is to first prove existence for
suitably regularized NS equations, which is possible using standard methods
of mathematical analysis, and to then prove that a regularized solution passes
as an
-weak solution for some
depending on the regularization, which tends
to zero with the regularization.

132

16 Existence of -Weak Solutions by G2

16.2 The Basic Energy Estimate for the Navier–Stokes
Equations

We start by deriving a basic stability estimate of energy type for the velocity
u of the NS equations (5.3), assuming for simplicity that f = 0. This is about
the only analytical a priori estimate known for the NS equations. We thus
formally assume existence of a (pointwise) solution (u, p), and derive a bound
for the velocity u in terms of given data.

Scalar multiplication of the momentum equation by u and integration with

respect to x gives

1
2

d

dt



Ω

|u|

2

dx + ν

3

i=1



Ω

|∇u

i

|

2

dx = 0,

because by partial integration (with boundary terms vanishing),



Ω

∇p · u dx = −



Ω

p

∇ · u dx = 0

and



Ω

(u

· ∇)u · u dx = −



Ω

(u

· ∇)u · u dx −



Ω

∇ · u|u|

2

dx

so that



Ω

(u

· ∇)u · u dx = 0.

(16.1)

Integrating next with respect to time, we obtain the following basic a priori
stability estimate for ˆ

t > 0 in terms of the L

2

-norm of the initial velocity u

0

:

E

ν

(u)

≡

1
2

u(·, ˆt )

2

+ D

ν

(u, ˆ

t ) =

1
2

u

0



2

,

(16.2)

where

D

ν

(u, ˆ

t ) = ν

3

i=1



ˆ

t

0

∇u

i



2

dt,

and where

 ·  denotes the L

2

(Ω)-norm. This estimate gives a bound on

the kinetic energy of the velocity with D

ν

(u, ˆ

t) representing the total dissipa-

tion from the viscosity of the fluid over the time interval [0, ˆ

t ]. We see that

the growth of this term with time corresponds to a decrease of the velocity
(momentum) of the flow (with f = 0).

The characteristic feature of a turbulent flow is that D

ν

(u, ˆ

t) is compara-

tively large, while in a laminar flow with ν small, D

ν

(u, ˆ

t) is small. With

D

ν

(u, ˆ

t)

∼ 1 in a turbulent flow and |∇u| uniformly distributed, we may

expect to have pointwise

|∇u

i

| ∼ ν

−1/2

.

(16.3)

16.3 Existence by G2

133

16.3 Existence by G2

To generate approximate weak solutions of the NS equations, we use a finite
element method of the form (assuming for simplicity f = 0): Find ˆ

U

≡ ˆ

U

h

∈

ˆ

V

h

, where ˆ

V

h

⊂ ˆ

V is a finite dimensional subspace of piecewise polynomial

functions defined on a computational mesh in space-time of mesh size h, such
that

((R( ˆ

U ), ˆ

v)) + ((hR( ˆ

U ), R(ˆ

v))) = 0,

∀ˆv ∈ ˆ

V

h

,

(16.4)

where R( ˆ

w)

≡ (R

1

( ˆ

w), R

2

(w)), ˆ

w = (w, r) and

R

1

( ˆ

w) = ˙

w + U

· ∇w + ∇r − ν∆w,

R

2

(w) =

∇ · w,

(16.5)

with element-wise definition of second order terms. We here interpret a con-
vection term ((U

· ∇w, v)) as

1
2

((U

· ∇w, v)) −

1
2

((U

· ∇v, w)

which is literally true if

∇ · U = 0. With this interpretation we will have

((U

· ∇U, U)) = 0, which corresponds to (16.1), even if the divergence of the

finite element velocity U does not vanish exactly. With this interpretation we
obtain choosing ˆ

v = ˆ

U in (16.4) (still assuming f = 0):

E

ν

(U ) + ((hR( ˆ

U ), R( ˆ

U ))) =

1
2

u

0



2

.

(16.6)

Fig. 16.1. Richard Courant (1888–1972) introduced the finite element method in
1922, in an existence proof of a version of the Riemann mapping theorem. Boris
Grigorievich Galerkin (1871–1945), Russian engineer who introduced the finite
element method as a computational tool.

The finite element method (16.4) is a stabilized Galerkin method (Fig. 16.1)

with the term ((R( ˆ

U ), v)) corresponding to Galerkin’s method and the term

134

16 Existence of -Weak Solutions by G2

((hR(ˆ

u), R(ˆ

v))) corresponding to a weighted residual least squares method

with stabilizing effect expressed in (16.6). We also refer to this method as
General Galerkin or G2, and we thus refer to ˆ

U as a G2-solution. The ex-

istence of a discrete solution ˆ

U

≡ ˆ

U

h

∈ V

h

follows by Brouwer’s fixed point

theorem combined with the stability estimate (16.6).

We now return to the main objective of this chapter of showing the exis-

tence of
-weak solutions to the NS equations. For all ˆ

v

∈ ˆ

V , we have with

ˆ

v

h

∈ ˆ

V

h

a standard interpolant of v satisfying

h

−1

(ˆ

v

− ˆv

h

)

 ≤ C

i

ˆv

ˆ

V

, using

also (16.4),

((R( ˆ

U ), ˆ

v)) = ((R( ˆ

U ), ˆ

v

− ˆv

h

))

− ((hR( ˆ

U ), R(ˆ

v

h

)))

≤ C

i

hR( ˆ

U )

ˆv

ˆ

V

+ M (U )

hR( ˆ

U )

ˆv

ˆ

V

,

(16.7)

where M (U ) is a pointwise bound of the velocity U (x, t), and C

i

≈ 1 is an

interpolation constant. It follows that the G2-solution ˆ

U is an
-weak solution

with

= (C

i

+ M (U ))

hR( ˆ

U )

 ≤

√

h(C

i

+ M (U ))

u

0

,

since from the energy stability estimate



√

hR( ˆ

U )

 ≤ u

0

.

Assuming now that M (U ) = M (U

h

) is bounded with h > 0, and letting

C

i

+ M (U )

≤ C, it follows that ˆ

U is an
-weak solution with
= C

√

h,

assuming

u

0

 ≤ 1. More generally, we may say that a G2 solution ˆ

U is an

-weak solution with
= C

hR( ˆ

U )

.

We have now demonstrated the existence of an
-weak solution to the

NS equations for any
, assuming that the maximum computed velocity is
bounded (or grows slower than h

−1/2

). More generally, we have shown that a

G2-solution ˆ

U is an
-weak solution with
= C

U

hR( ˆ

U )

 with C

U

= C

i

+

M (U ). Computing ˆ

U , we can compute
= C

U

hR( ˆ

U )

 and thus determine

the corresponding
.

We conclude that coming up with
-weak solutions to the NS equations

is easy, if we use G2 and a computer (and find that C

U

grows slower than

h

−1/2

).

We now turn to the question of estimation of the error in output of G2-

solutions, which of course as above will bring in the corresponding stability
factor.

Remark. In estimating above ((R( ˆ

U ), ˆ

v

−ˆv

h

)) we did not properly account for

the diffusion term ((ν

∇U, ∇(v−v

h

))). Doing so would introduce an additional

term which most easily can be estimated by a term of the form C

√

ν

ˆv

ˆ

V

,

and to bound this term as above we would need that ν

≤ h. Since ν often is

smaller than 10

−4

for the problems we focus on in this book, this would not

be restrictive in most cases. For larger ν we can turn the argument around in
a different way, but we do not here enter into details.

16.4 A Posteriori Output Error Estimate for G2

135

16.4 A Posteriori Output Error Estimate for G2

We now let ˆ

u be an
-weak solution of the NS equations and let ˆ

U be a G2-

solution, which we just showed can be viewed to be an

G2

-weak solution,

with

G2

= C

U

hR( ˆ

U )

 >>
.

As above we get the following a posteriori error estimate for a mean-value

output given by a function ˆ

ψ:

|M(ˆu) − M( ˆ

U )

| ≤ (
+ C

U

hR( ˆ

U )

)S



G2

( ˆ

ψ),

(16.8)

where S



G2

( ˆ

ψ) is the corresponding stability factor defined as above. Obviously

the size of the stability factor S



G2

( ˆ

ψ) is crucial for computability: the stopping

criterion is evidently (assuming
small):

C

U

hR( ˆ

U )

S



G2

( ˆ

ψ)

≤ T OL,

where T OL > 0 is a tolerance. If S



G2

( ˆ

ψ) is too large, or T OL is too small, then

we may not be able to reach the stopping criterion with available computing
power, and the computability is out of reach.