18

A Convection-Diffusion Model Problem

How can it be that even if everything I do is pointwise wrong (according to
my critics), yet my mean value comes out right? (Oscar Wilde)

18.1 Introduction

We discuss some basic aspects of G2 in the setting of a convection-diffusion
model problem. We first comment on the fact that the residual R(ˆ

u) of an

-weak solution ˆ

u necessarily is pointwise large where the flow is turbulent and

not fully resolved. In fact, a turbulent flow is characterized by the fact that
the stabilization term is not small and thus the residual large pointwise. We
then show that the least squares stabilization of G2 introduces an artificial
viscosity acting as a turbulent diffusion on smallest scales only and therefore
does not degrade the accuracy of mean value outputs.

18.2 Pointwise vs Mean Value Residuals

We have noticed above that even though the residual R(ˆ

u) of an
-weak solu-

tion ˆ

u of the NS equations is not small pointwise, its effect on a mean value

output M (ˆ

u) may be small. We will now discover the same phenomenon in

the following scalar linear constant coefficient stationary convection-diffusion-
reaction model problem with small viscosity ν:

u

,1

+ u

− ν∆u = f in Ω, u = 0 on Γ,

(18.1)

where Ω = (0, 1)

2

with boundary Γ , u

,1

=

∂u

∂x

1

and f is a given (smooth)

function. The solution u = u(x) in general has an outflow boundary layer at
x

1

= 1 of width

∼ ν, and characteristic layers at x

2

= 0 and x

2

= 1 of width

∼

√

ν. With an oscillating inflow value of u(x) at x

1

= 0, the characteristic

layers may fill Ω.

148

18 A Convection-Diffusion Model Problem

Let now V

h

be the standard finite element space of continuous piecewise

linear functions on a triangulation of Ω of mesh size h vanishing on Γ , and
let U

∈ V

h

be a G2 solution defined by

(U

,1

+ U, v + hv

,1

) = (f, v + hv

,1

)

(18.2)

where (

·, ·) is the L

2

(Ω)-norm, and we assume that ν << h so that the ν-

term can be omitted in G2. Further, the stabilizing term was simplified from
h(v

,1

+ v) to hv

,1

, as in the streamline diffusion method [35]. Choosing here

v = U gives the following basic energy estimate for U :

U

2

+



√

hU

,1



2

≤ (1 + h)f

2

≈ 1

where

 ·  is the L

2

(Ω)-norm and we assume

f = 1. We notice that in the

case the exact solution u has layers, the stabilizing term



√

hU

,1



2

will not be

small, because U

,1

∼ h

−1

in an outflow layer of width

∼ h, and U

,1

∼ h

−1/2

in characteristic layers of width h

1/2

.

Now, if ν << h, then

R(U )

≈ U

,1

+ U

− f

and thus by choosing v = U in (18.2)

−(R(U), U) ≈ 

√

hU

,1



2

>> 0,

which shows that R(U ) cannot be pointwise small everywhere in Ω: We will
argue below that R(U )

∼ h

−1

in outflow layers and R(U )

∼ 1 in characteristic

layers. Note that



√

hU

,1



2

not small signifies the presence of unresolved layers

where R(U ) is not small, which mimics the fact that in the NS equations the
stabilizing term and residual are not small in unresolved (turbulent) regions.

Now, if we in the model problem take as output M (u) = (u, ψ), where ψ

vanishes in the layers, that is we consider only output away from the numerical
layers, then it follows by the analysis of G2 in [70], that if u is smooth outside
layers then

|M(u) − M(U)| ∼ h

3/2

,

which shows that the error in certain outputs may be small even though the
residual R(U ) is large pointwise in certain parts of the domain. Of course,
in the model problem this is fully understandable because by the nature of
the convection in the positive x

1

-direction and the smallness of the diffusion

coefficients, effects in boundary layers are not propagated into the domain.
Alternatively, we may as above in the case of the NS equations bound the
output error in terms of

R(U)

H

−1

(Ω)

∼ hR(U),

which may be small (

∼

√

h), even though R(U ) is not small everywhere.

18.3 Artificial Viscosity From Least Squares Stabilization

149

To understand more precisely why the residual R(U ) cannot be small in an

outflow layer, we note that the exact solution u with f smooth there satisfies

u

,1

+ u

− f = ν∆u ∼

1

ν

(18.3)

if u varies between 0 and 1 in the layer, so that R(u) = 0 results from cancel-
lation of the two terms u

,1

+ u

1

−f and ν∆u both ∼ 1/ν. In the numerics this

cancellation cannot be realized if h >> ν, and the result is that R(u)

≈ 1/h

in the numerical outflow layer: Roughly speaking, we have in an outflow layer

U

,1

+ U

1

− f ∼ hU

,11

∼

1

h

,

which is incompatible with (18.3) if h >> ν, and thus necessarily R(U )

≈ 1/h

in an outflow layer. A similar argument shows that we may have R(U )

≈ 1 in

a characteristic layer. We may say that the fact that R(U ) cannot be small
in layers, is a necessary consequence of the under resolution with h >> ν,
which makes it impossible to numerically capture the cancellation of non-
small viscous and non-viscous terms present in the continuous problem.

We sum up by noting that the under resolution with h >> ν makes it

impossible for R(U ) to be pointwise small everywhere, while the fact that
R(U)

H

−1

(Ω)

is small opens for the possibility that the error in a mean

value output is small, if the dual solution is not too large.

18.3 Artificial Viscosity From Least Squares
Stabilization

The least squares stabilization in (18.2) effectively introduces the term

(U

,1

+ U

− f, hv

,1

)

which involves the artificial viscosity (U

,1

, hv

,1

). The stabilizing term is ob-

viously small where U

,1

+ U

1

− f is small, that is outside layers where the

solution is smooth. The net effect of the least squares stabilization thus is in-
creased viscosity only in regions where the solution is non-smooth. Similarly
the G2 stabilization in NS has the effect of a turbulent diffusion acting only
on the smallest scales of the flow.

As a comparison we note that simply adding an artificial viscosity term

(U

,1

, hv

,1

) without the compensating terms in (U

,1

+ U

− f, hv

,1

), introduces

a perturbation of order h also to smooth parts of the flow, which significantly
degrades the accuracy. Adding artificial viscosity in the form (U

,1

, hv

,1

) corre-

sponds to the simplest version of the classical Smagorinsky turbulence model
in NS.

The least squares stabilization in G2 thus may be viewed as a smart

Smagorinsky model (see Chapter 26), effectively introducing diffusion only

150

18 A Convection-Diffusion Model Problem

on the smallest scales of the mesh. The rationale is then that the actual size
of the smallest scale of the diffusion is insignificant for certain mean value
outputs, and thus that certain aspects of turbulent flow can be captured on
computational scales which are (much) coarser than the actual physical scales.
This reflects that our World would look the same even if the “fluid particles”
were much bigger than particles on atomic scales.