3

THE CONVECTION–DIFFUSION EQUATION

We next consider the convection–di!usion equation

!!"2u + "w · "u = f, (3.1)

where !> 0. This equation arises in numerous models of flows and other physicalphenomena. The unknown function u may represent the concentration of a pol-lutant being transported (or “convected”) along a stream moving at velocity "wand also subject to di!usive e!ects. Alternatively, it may represent the temper-ature of a fluid moving along a heated wall, or the concentration of electronsin models of semiconductor devices. This equation is also a fundamental sub-problem for models of incompressible flow, considered in Chapter 7, where "uis a vector-valued function representing flow velocity and ! is a viscosity para-meter. Typically, di!usion is a less significant physical e!ect than convection: ona windy day the smoke from a chimney moves in the direction of the wind andany spreading due to molecular di!usion is small. This implies that, for mostpractical problems, ! # |"w|. This chapter is concerned with the properties offinite element discretization of the convection–di!usion equation, and Chapter 4with e!ective algorithms for solving the discrete linear equation systems thatarise from the discretization process.

The boundary value problem that is considered is equation (3.1) posed ona two-dimensional or three-dimensional domain #, together with boundaryconditions on $# = $#D $ $#N given by (1.14), that is,

u = gD on $#D,$u

$n= gN on $#N . (3.2)

We will assume, as is commonly the case, that the flow characterized by "w isincompressible, that is, div "w = 0. The domain boundary $# will be subdividedaccording to its relation with the velocity field "w: if "n denotes the outward-pointing normal to the boundary, then

$#+ = {x % $# | "w · "n > 0}, the outflow boundary,

$#0 = {x % $# | "w · "n = 0}, the characteristic boundary,

$#! = {x % $# | "w · "n < 0}, the inflow boundary.

The presence of the first-order convection term "w · "u gives theconvection–di!usion equation a decidedly di!erent character from that of thePoisson 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 ofthe domain tends to be close to the solution, u, of the hyperbolic equation

"w · "u = f. (3.3)

Let us first briefly consider this alternative equation, which will be referred to asthe reduced problem. The di!erential operator of (3.3) is of lower order than thatof (3.1), and u generally cannot satisfy all the boundary conditions imposed on u.To see this, consider the streamlines or characteristic curves associated with "was 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 characterizationd"c/ds = "w implies that, for a fixed streamline, the solution to the reducedproblem satisfies the ordinary di!erential equation

dds

[u("c(s))] = f("c(s)). (3.4)

Equivalently, if f = 0, the reduced solution is constant along streamlines. Supposefurther that the parameterization is such that "c(s0) lies on the inflow boundary$#! for some s0. If u("c(s0)) is used to specify an initial condition for the dif-ferential equation (3.4), and if, in addition, for some s1 > s0, "c(s1) intersectsanother point on $# (say on the outflow boundary $#+), then the boundaryvalue u("c(s1)) is determined by solving (3.4). This value need not have any rela-tion 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 $#!, thecharacterization (3.4) implies that u will have a discontinuity propagating into# along the streamline that originates at the point of discontinuity on the inflowboundary.

Because of these phenomena, it often happens that the solution u to theconvection–di!usion equation has a steep gradient in a portion of the domain.For example, u may be close to u in most of #, but along a streamline goingto an outflow boundary where u and u di!er, u will exhibit a steep gradient inorder to satisfy the boundary condition. (The following one-dimensional problemo!ers a lot of insight here: given ! and w > 0, find u(x) such that !!u"" +wu" = 1for x % (0, L), with u(0) = 0, u(L) = 0; see Problem 3.1.) In such a situation, theproblem defined by (3.1) and boundary conditions (3.2) is said to be singularlyperturbed, and the solution u has an exponential boundary layer. The di!usion 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 whereu is smoother than u. For example, for a discontinuous boundary conditionon $#! as discussed above, the di!usion term in (3.1) leads to a smoothingof the discontinuity inside #. In this instance the solution u is continuous butrapidly varying across an internal layer that follows the streamline emanatingfrom the discontinuity on the inflow boundary. The presence of layers of bothtypes makes it di"cult to construct accurate discrete approximations in caseswhen convection is dominant.

Equally significant, when "w &= "0, the boundary value problem (3.1)–(3.2) isnot self-adjoint. (This means that

!!(Lu)v &=

!! u (Lv) , where the di!erential

operator of (3.1) is denoted by L = !!"2 + "w · ". ) As a result, the coe"cientmatrix derived from discretization is invariably nonsymmetric — in contrast tothe Poisson problem where the coe"cient matrix is always symmetric positive-semidefinite. Non-symmetry in turn a!ects the choice and performance of iterat-ive solution algorithms for solving the discrete problems, and di!erent techniquesfrom those discussed in Chapter 2 must be used to achieve e!ective performance.

In discussing convection–di!usion equations, it is useful to have a quantitativemeasure of the relative contributions of convection and di!usion. This can bedone by normalizing equation (3.1) with respect to the size of the domain andthe magnitude of the velocity. Thus, as above, let L denote a characterizinglength scale for the domain #; for example, L can be the Poincare constant ofLemma 1.2. In addition, let the velocity "w be specified as "w = W"w# where W is apositive constant and |"w#| is normalized to have value unity in some measure | · |.If points in # are denoted by "x, then "% = "x/L denotes elements of a normalizeddomain. With u#("%) = u(L"%) on this domain, (3.1) can be rewritten as

!"2u# +"

WL!

#"w# · "u# =

L2

!f, (3.5)

and the relative contributions of convection and di!usion can be encapsulatedin the Peclet number:

P :=WL!

. (3.6)

If P ' 1, equation (3.5) is di!usion-dominated and relatively benign. In contrast,the construction of accurate approximations and the design of e!ective solversin the convection-dominated case, (3.5) with P ( 1, will be shown to be fraughtwith di"culty.

3.1 Reference problems

Here and subsequently, we will refer to the velocity vector "w as the wind. Severalexamples of two-dimensional convection–di!usion problems will be used to illus-trate the e!ect 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 # ) # = (!1, 1) * (!1, 1), with wind of order unity+"w+$ = O(1) and zero source term f = 0. Since the Peclet number is inverselyproportional to ! the problems are convection-dominated if ! # 1. The qualityand accuracy of discretizations of these problems will be discussed in Sections 3.3and 3.4, and special issues relating to solution algorithms will be considered inChapter 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 theboundary $# 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 streamlinesare given by the vertical lines c(s) = (&, s) where & % (!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 uis very close to that of the reduced problem u ) x except near the outflowboundary y = 1, where it is zero.

–1 –0.5 0 0.5 1–1

–0.5

0

0.5

1

u = x

u = 0

u ! –1 u ! 1

–1 0 1 –1

0

1

–1

–0.5

0

0.5

1

y

x

Fig. 3.2. Contour plot (left) and three-dimensional surface plot (right) of anaccurate 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 boundarylayer. In this example, the layer is determined by the function e(1!y)/" and haswidth proportional to !. Figure 3.2 shows a contour plot and three-dimensionalrendering of the solution for != 1/200. Using asymptotic expansions (seeEckhaus [53]), it can be shown in general that boundary layers arisingfrom “hard” Dirichlet conditions on the outflow boundary can be representedusing 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 bound-ary layers

In this example the wind is vertical "w = (0, 1 + (x + 1)2/4) but increases instrength from left to right. Dirichlet boundary values apply on the inflow andcharacteristic boundary segments; u is set to unity on the inflow boundary, anddecreases 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 thereis no exponential boundary layer in this case. The fact that the reduced solutionu ) 1 is incompatible with the specified values on the characteristic boundary,generates characteristic layers on each side. These layers are typical of so-calledshear layers that commonly arise in fluid flow models. The width of shear layersis 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 inFigure 3.2. The reason that the layer on the right is sharper than the layer onthe left is that the wind is twice as strong along the associated boundary.

u = 1

" u / " y = 0

–10

1

–1

0

10

0.5

1

Fig. 3.3. Contour plot (left) and three-dimensional surface plot (right) of anaccurate 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 ofvertical, downstream boundary layer and interior layer.

In this example, the wind is a constant vector "w =$! sin #

6 , cos #6

%. Dirichlet

boundary conditions are imposed everywhere on $#, with values either zero orunity 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 $#,[x,!1] $ [1, y], and the reduced problem solution u is constant along thestreamlines

&(x, y)

''' 12y +&

32 x = constant

(, (3.8)

with values determined by the inflow boundary condition. The discontinuity ofthe boundary condition causes u to be a discontinuous function with the valueu = 0 to the left of the streamline y +

-3x = !1 and the value u = 1 to the

right. The di!usion term present in (3.1) causes this discontinuity to be smeared,producing an internal layer of width O(

-!). There is also an exponential bound-

ary layer near the top boundary y = 1, where the value of u drops rapidly fromu , 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 factthat in two dimensions, incompressibility via div "w = 0 implies that

"w ="

!$'

$y,$'

$x

#T

, (3.9)

–1 –0.5 0 0.5 1–1

–0.5

0

0.5

1

u = 0 u = 1

u = 0

u = 0 u = 1

–1 0 1–1

0

1

0

0.5

1

xy

Fig. 3.4. Contour plot (left) and three-dimensional surface plot (right) of anaccurate finite element solution of Example 3.1.3, for ! = 1/200.

REFERENCE PROBLEMS 119

where '(x, y) is an associated stream function. Defining the level curves of ' asthe set of points (x, y) % # for which

'(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 thestreamlines for two-dimensional problems. This alternative characterization canalso 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 inflowboundary segment so that the corresponding reduced problem (3.3) does nothave a uniquely defined solution.

3.1.4 Example: Zero source term, recirculating wind, characteristic boundarylayers.

This example is known as the double-glazing problem: it is a simple modelfor the temperature distribution in a cavity with an external wall that is “hot”.The wind "w = (2y(1 ! x2),!2x(1 ! y2)) determines a recirculating flow withstreamlines

{(x, y) | (1 ! x2)(1 ! y2) = constant}.

All boundaries are of characteristic type. Dirichlet boundary conditions areimposed everywhere on $#, and there are discontinuities at the two cornersof the hot wall, x = 1, y = ±1. These discontinuities lead to boundary layersnear these corners, as shown in Figure 3.5. Although these layers are compar-able in width to those in Figure 3.3, it is emphasized that their structure is notaccessible via asymptotic techniques, unlike the layers in the first three examples.

–1 –0.5 0 0.5 1–1

–0.5

0

0.5

1

u = 0

u = 0

u = 0 u = 1

–10

1

–1

0

10

0.5

1

xy

Fig. 3.5. Contour plot (left) and three-dimensional surface plot (right) of anaccurate finite element solution of Example 3.1.4, for ! = 1/200.