3
THE CONVECTION–DIFFUSION EQUATION
We next consider the convection–diffusion equation
−!∇2 u + w
" · ∇u = f,
(3.1)
where ! > 0. This equation arises in numerous models of flows and other physical
phenomena. The unknown function u may represent the concentration of a pollutant being transported (or “convected”) along a stream moving at velocity w
"
and also subject to diffusive effects. Alternatively, it may represent the temperature of a fluid moving along a heated wall, or the concentration of electrons
in models of semiconductor devices. This equation is also a fundamental subproblem for models of incompressible flow, considered in Chapter 7, where "u
is a vector-valued function representing flow velocity and ! is a viscosity parameter. Typically, diffusion is a less significant physical effect than convection: on
a windy day the smoke from a chimney moves in the direction of the wind and
any spreading due to molecular diffusion is small. This implies that, for most
practical problems, ! # |w|.
" This chapter is concerned with the properties of
finite element discretization of the convection–diffusion equation, and Chapter 4
with effective algorithms for solving the discrete linear equation systems that
arise from the discretization process.
The boundary value problem that is considered is equation (3.1) posed on
a two-dimensional or three-dimensional domain Ω, together with boundary
conditions on ∂Ω = ∂ΩD ∪ ∂ΩN given by (1.14), that is,
∂u
= gN on ∂ΩN .
∂n
u = gD on ∂ΩD ,
(3.2)
We will assume, as is commonly the case, that the flow characterized by w
" is
incompressible, that is, div w
" = 0. The domain boundary ∂Ω will be subdivided
according to its relation with the velocity field w:
" if "n denotes the outwardpointing normal to the boundary, then
∂Ω+ = {x ∈ ∂Ω | w
" · "n > 0},
the outflow boundary,
" · "n < 0},
∂Ω− = {x ∈ ∂Ω | w
the inflow boundary.
" · "n = 0},
∂Ω0 = {x ∈ ∂Ω | w
the characteristic boundary,
The presence of the first-order convection term w
" · ∇u gives the
convection–diffusion equation a decidedly different character from that of the
Poisson equation. Under the assumption that !/(|w|L)
"
is small, where L is a
113
114
THE CONVECTION–DIFFUSION EQUATION
characteristic length scale associated with (3.1), the solution to (3.1) in most of
the domain tends to be close to the solution, û, of the hyperbolic equation
w
" · ∇û = f.
(3.3)
Let us first briefly consider this alternative equation, which will be referred to as
the reduced problem. The differential operator of (3.3) is of lower order than that
of (3.1), and û generally cannot satisfy all the boundary conditions imposed on u.
To see this, consider the streamlines or characteristic curves associated with w
"
as illustrated in Figure 3.1. These are defined to be the parameterized curves
"c(s) that have tangent vector w("
" c(s)) at every point on "c. The characterization
d"c/ds = w
" implies that, for a fixed streamline, the solution to the reduced
problem satisfies the ordinary differential equation
d
[û("c(s))] = f ("c(s)).
ds
(3.4)
Equivalently, if f = 0, the reduced solution is constant along streamlines. Suppose
further that the parameterization is such that "c(s0 ) lies on the inflow boundary
∂Ω− for some s0 . If u("c(s0 )) is used to specify an initial condition for the differential equation (3.4), and if, in addition, for some s1 > s0 , "c(s1 ) intersects
another point on ∂Ω (say on the outflow boundary ∂Ω+ ), then the boundary
value û("c(s1 )) is determined by solving (3.4). This value need not have any relation to the corresponding value taken on by u, which is determined by (3.2).
Notice also that in the case of a discontinuous boundary condition on ∂Ω− , the
characterization (3.4) implies that û will have a discontinuity propagating into
Ω along the streamline that originates at the point of discontinuity on the inflow
boundary.
Because of these phenomena, it often happens that the solution u to the
convection–diffusion equation has a steep gradient in a portion of the domain.
For example, u may be close to û in most of Ω, but along a streamline going
to an outflow boundary where u and û differ, u will exhibit a steep gradient in
order to satisfy the boundary condition. (The following one-dimensional problem
offers a lot of insight here: given ! and w > 0, find u(x) such that −!u"" +wu" = 1
for x ∈ (0, L), with u(0) = 0, u(L) = 0; see Problem 3.1.) In such a situation, the
problem defined by (3.1) and boundary conditions (3.2) is said to be singularly
perturbed, and the solution u has an exponential boundary layer. The diffusion in
Fig. 3.1. Streamline "c(s) associated with vector field w.
"
REFERENCE PROBLEMS
115
equation (3.1) may also lead to steep gradients transverse to streamlines where
u is smoother than û. For example, for a discontinuous boundary condition
on ∂Ω− as discussed above, the diffusion term in (3.1) leads to a smoothing
of the discontinuity inside Ω. In this instance the solution u is continuous but
rapidly varying across an internal layer that follows the streamline emanating
from the discontinuity on the inflow boundary. The presence of layers of both
types makes it difficult to construct accurate discrete approximations in cases
when convection is dominant.
Equally significant, when w
" &= "0,
value problem (3.1)–(3.2) is
! the boundary
!
not self-adjoint. (This means that Ω (Lu)v &= Ω u (Lv) , where the differential
" · ∇. ) As a result, the coefficient
operator of (3.1) is denoted by L = −!∇2 + w
matrix derived from discretization is invariably nonsymmetric — in contrast to
the Poisson problem where the coefficient matrix is always symmetric positivesemidefinite. Non-symmetry in turn affects the choice and performance of iterative solution algorithms for solving the discrete problems, and different techniques
from those discussed in Chapter 2 must be used to achieve effective performance.
In discussing convection–diffusion equations, it is useful to have a quantitative
measure of the relative contributions of convection and diffusion. This can be
done by normalizing equation (3.1) with respect to the size of the domain and
the magnitude of the velocity. Thus, as above, let L denote a characterizing
length scale for the domain Ω; for example, L can be the Poincaré constant of
Lemma 1.2. In addition, let the velocity w
" be specified as w
" = Ww
" ∗ where W is a
positive constant and |w
" ∗ | is normalized to have value unity in some measure | · |.
If points in Ω are denoted by "x, then ξ" = "x/L denotes elements of a normalized
" = u(Lξ)
" on this domain, (3.1) can be rewritten as
domain. With u∗ (ξ)
"
#
WL
L2
2
f,
(3.5)
−∇ u∗ +
w
" ∗ · ∇u∗ =
!
!
and the relative contributions of convection and diffusion can be encapsulated
in the Peclet number:
P :=
WL
.
!
(3.6)
If P ≤ 1, equation (3.5) is diffusion-dominated and relatively benign. In contrast,
the construction of accurate approximations and the design of effective solvers
in the convection-dominated case, (3.5) with P ( 1, will be shown to be fraught
with difficulty.
3.1
Reference problems
Here and subsequently, we will refer to the velocity vector w
" as the wind. Several
examples of two-dimensional convection–diffusion problems will be used to illustrate the effect of the wind direction and strength on properties of solutions,
and on the quality of finite element discretizations. The problems are all posed
116
THE CONVECTION–DIFFUSION EQUATION
on the square domain Ω ≡ Ω = (−1, 1) × (−1, 1), with wind of order unity
+w+
" ∞ = O(1) and zero source term f = 0. Since the Peclet number is inversely
proportional to ! the problems are convection-dominated if ! # 1. The quality
and accuracy of discretizations of these problems will be discussed in Sections 3.3
and 3.4, and special issues relating to solution algorithms will be considered in
Chapter 4.
3.1.1 Example: Analytic solution, zero source term, constant vertical wind,
exponential boundary layer.
The function
u(x, y) = x
"
1 − e(y−1)/"
1 − e−2/"
#
(3.7)
satisfies equation (3.1) with w
" = (0, 1) and f = 0. Dirichlet conditions on the
boundary ∂Ω are determined by (3.7) and satisfy
u(x, −1) = x,
u(x, 1) = 0,
u(−1, y) ≈ −1, u(1, y) ≈ 1,
where the latter two approximations hold except near y = 1. The streamlines
are given by the vertical lines c(s) = (α, s) where α ∈ (−1, 1) is constant,
giving a flow in the vertical direction. On the characteristic boundaries x = ±1,
the boundary values vary dramatically near y = 1, changing from (essentially)
−1 to 0 on the left and from +1 to 0 on the right. For small !, the solution u
is very close to that of the reduced problem û ≡ x except near the outflow
boundary y = 1, where it is zero.
u=0
1
0.5
1
u≈1
0.5
u ≈ –1
0
0
–0.5
1
–0.5
–1
–1
–0.5
0
u=x
0.5
1
–1
–1
0
0
1
–1
y
x
Fig. 3.2. Contour plot (left) and three-dimensional surface plot (right) of an
accurate finite element solution of Example 3.1.1, for ! = 1/200.
REFERENCE PROBLEMS
117
The dramatic change in the value of u near y = 1 constitutes a boundary
layer. In this example, the layer is determined by the function e(1−y)/" and has
width proportional to !. Figure 3.2 shows a contour plot and three-dimensional
rendering of the solution for ! = 1/200. Using asymptotic expansions (see
Eckhaus [53]), it can be shown in general that boundary layers arising
from “hard” Dirichlet conditions on the outflow boundary can be represented
using exponential functions in local coordinates. Following Roos et al.
[159, Section III.1.3], we refer to such layers as exponential boundary layers.
For +w+
" ∞ &= 1, they have width inversely proportional to the Peclet number.
3.1.2 Example: Zero source term, variable vertical wind, characteristic boundary layers
In this example the wind is vertical w
" = (0, 1 + (x + 1)2 /4) but increases in
strength from left to right. Dirichlet boundary values apply on the inflow and
characteristic boundary segments; u is set to unity on the inflow boundary, and
decreases to zero quadratically on the right wall, and cubically on the left wall,
see Figure 3.3. A zero Neumann condition on the top boundary ensures that there
is no exponential boundary layer in this case. The fact that the reduced solution
û ≡ 1 is incompatible with the specified values on the characteristic boundary,
generates characteristic layers on each side. These layers are typical of so-called
shear layers that commonly
arise in fluid flow models. The width of shear layers
√
is proportional to ! rather than !, so they are less intimidating than exponential
layers — this can be seen by comparing the solution in Figure 3.3 with that in
Figure 3.2. The reason that the layer on the right is sharper than the layer on
the left is that the wind is twice as strong along the associated boundary.
∂u/∂y=0
1
0.5
0
1
1
0
u=1
–1
0
–1
Fig. 3.3. Contour plot (left) and three-dimensional surface plot (right) of an
accurate finite element solution of Example 3.1.2, for ! = 1/200.
118
THE CONVECTION–DIFFUSION EQUATION
3.1.3 Example: Zero source term, constant wind at a 30◦ angle to the left of
vertical, downstream boundary layer and interior layer.
%
$
In this example, the wind is a constant vector w
" = − sin π6 , cos π6 . Dirichlet
boundary conditions are imposed everywhere on ∂Ω, with values either zero or
unity with a jump discontinuity at the point (0, −1) as illustrated in Figure 3.4.
The inflow boundary is composed of the bottom and right portions of ∂Ω,
[x, −1] ∪ [1, y], and the reduced problem solution û is constant along the
streamlines
'
&
'
(x, y) ' 12 y +
√
3
2 x
(
= constant ,
(3.8)
with values determined by the inflow boundary condition. The discontinuity of
the boundary condition causes û to be a√discontinuous function with the value
û = 0 to the left of the streamline y + 3x = −1 and the value û = 1 to the
right. The diffusion term present in (3.1)√causes this discontinuity to be smeared,
producing an internal layer of width O( !). There is also an exponential boundary layer near the top boundary y = 1, where the value of u drops rapidly from
u ≈ 1 to u = 0. Figure 3.4 shows contour and surface plots of the solution for
! = 1/200.
An alternative characterization of streamlines may be obtained from the fact
that in two dimensions, incompressibility via div w
" = 0 implies that
w
"=
"
∂ψ ∂ψ
−
,
∂y ∂x
#T
(3.9)
,
u=0
1
1
0.5
u=0
u=1
0.5
0
0
–0.5
–1
–1
1
0
–0.5
u=0
0
0.5
u=1
1
y
–1
–1
x
0
1
Fig. 3.4. Contour plot (left) and three-dimensional surface plot (right) of an
accurate finite element solution of Example 3.1.3, for ! = 1/200.
REFERENCE PROBLEMS
119
where ψ(x, y) is an associated stream function. Defining the level curves of ψ as
the set of points (x, y) ∈ Ω for which
ψ(x, y) = constant,
(3.10)
it can be shown that the streamlines in (3.8) are parallel to these level curves,
see Problem 3.2. The upshot is that (3.10) can be taken as the definition of the
streamlines for two-dimensional problems. This alternative characterization can
also be applied in cases when the basic definition is not applicable. It is used,
for example, in the following reference problem, which does not have an inflow
boundary segment so that the corresponding reduced problem (3.3) does not
have a uniquely defined solution.
3.1.4 Example: Zero source term, recirculating wind, characteristic boundary
layers.
This example is known as the double-glazing problem: it is a simple model
for the temperature distribution in a cavity with an external wall that is “hot”.
The wind w
" = (2y(1 − x2 ), −2x(1 − y 2 )) determines a recirculating flow with
streamlines
{(x, y) | (1 − x2 )(1 − y 2 ) = constant}.
All boundaries are of characteristic type. Dirichlet boundary conditions are
imposed everywhere on ∂Ω, and there are discontinuities at the two corners
of the hot wall, x = 1, y = ±1. These discontinuities lead to boundary layers
near these corners, as shown in Figure 3.5. Although these layers are comparable in width to those in Figure 3.3, it is emphasized that their structure is not
accessible via asymptotic techniques, unlike the layers in the first three examples.
u=0
1
1
0.5
u=0
u = 1 0.5
0
0
1
–0.5
–1
–1
1
0
–0.5
0
u=0
0.5
1
y
–1
0
–1
x
Fig. 3.5. Contour plot (left) and three-dimensional surface plot (right) of an
accurate finite element solution of Example 3.1.4, for ! = 1/200.