THE CONVECTION DIFFUSION EQUATION

3 THE CONVECTION DIFFUSION EQUATION We next consider the convection diffusion equation ɛ 2 u + w u = f, (3.) where ɛ>. 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 ɛ 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, ɛ 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.) posed on a two-dimensional or three-dimensional domain Ω, together with boundary conditions on Ω = Ω D Ω N given by (.4), that is, u = g D on Ω D, u n = g N on Ω N. (3.2) We will assume, as is commonly the case, that the flow characterized by w is incompressible, that is, div w =. 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 >}, Ω = {x Ω w n =}, Ω = {x Ω w n <}, the outflow boundary, the characteristic boundary, the inflow 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 ɛ/( w L) is small, where L is a 3

4 THE CONVECTION DIFFUSION EQUATION characteristic length scale associated with (3.), the solution to (3.) 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.), 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.. 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)). (3.4) ds Equivalently, if f =, the reduced solution is constant along streamlines. Suppose further that the parameterization is such that c(s ) lies on the inflow boundary Ω for some s. If u( c(s )) is used to specify an initial condition for the differential equation (3.4), and if, in addition, for some s >s, c(s ) intersects another point on Ω (say on the outflow boundary Ω + ), then the boundary value û( c(s )) 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 ɛ and w>, find u(x) such that ɛu +wu = for x (,L), with u() =,u(l) = ; see Problem 3..) In such a situation, the problem defined by (3.) 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.. Streamline c(s) associated with vector field w.

REFERENCE PROBLEMS 5 equation (3.) 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.) 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, the boundary value problem (3.) (3.2) is not self-adjoint. (This means that Ω (Lu)v u (Lv), where the differential Ω operator of (3.) is denoted by L = ɛ 2 + w. ) As a result, the coefficient 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.) 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.2. In addition, let the velocity w be specified as w =W w 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 domain. With u ( ξ)=u(l ξ) on this domain, (3.) can be rewritten as ( ) WL 2 u + ɛ w u = L2 f, (3.5) ɛ and the relative contributions of convection and diffusion can be encapsulated in the Peclet number: P := WL ɛ. (3.6) If P, 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, will be shown to be fraught with difficulty. 3. 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

6 THE CONVECTION DIFFUSION EQUATION on the square domain Ω Ω =(, ) (, ), with wind of order unity w = O() and zero source term f =. Since the Peclet number is inversely proportional to ɛ the problems are convection-dominated if ɛ. 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.. Example: Analytic solution, zero source term, constant vertical wind, exponential boundary layer. The function ( ) e (y )/ɛ u(x, y) =x e 2/ɛ (3.7) satisfies equation (3.) with w =(, ) and f =. Dirichlet conditions on the boundary Ω are determined by (3.7) and satisfy u(x, ) = x, u(x, ) =, u(,y), u(,y), where the latter two approximations hold except near y =. The streamlines are given by the vertical lines c(s) = (α, s) where α (, ) is constant, giving a flow in the vertical direction. On the characteristic boundaries x = ±, the boundary values vary dramatically near y =, changing from (essentially) to on the left and from + to on the right. For small ɛ, the solution u is very close to that of the reduced problem û x except near the outflow boundary y =, where it is zero. u=.5 u u.5.5.5.5 u=x.5 x y Fig. 3.2. Contour plot (left) and three-dimensional surface plot (right) of an accurate finite element solution of Example 3.., for ɛ = /2.

REFERENCE PROBLEMS 7 The dramatic change in the value of u near y = constitutes a boundary layer. In this example, the layer is determined by the function e ( y)/ɛ and has width proportional to ɛ. Figure 3.2 shows a contour plot and three-dimensional rendering of the solution for ɛ =/2. 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. [59, Section III..3], we refer to such layers as exponential boundary layers. For w, they have width inversely proportional to the Peclet number. 3..2 Example: Zero source term, variable vertical wind, characteristic boundary layers In this example the wind is vertical w = (, +(x + ) 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 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 ɛ rather than ɛ, 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 =.5 u= Fig. 3.3. Contour plot (left) and three-dimensional surface plot (right) of an accurate finite element solution of Example 3..2, for ɛ = /2.

8 THE CONVECTION DIFFUSION EQUATION 3..3 Example: Zero source term, constant wind at a 3 angle to the left of vertical, downstream boundary layer and interior layer. In this example, the wind is a constant vector w = ( sin π 6, cos π 6 ). Dirichlet boundary conditions are imposed everywhere on Ω, with values either zero or unity with a jump discontinuity at the point (, ) as illustrated in Figure 3.4. The inflow boundary is composed of the bottom and right portions of Ω, [x, ] [, y], and the reduced problem solution û is constant along the streamlines { (x, y) 2 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 = to the left of the streamline y + 3x = and the value û = to the right. The diffusion term present in (3.) causes this discontinuity to be smeared, producing an internal layer of width O( ɛ). There is also an exponential boundary layer near the top boundary y =, where the value of u drops rapidly from u tou =. Figure 3.4 shows contour and surface plots of the solution for ɛ =/2. An alternative characterization of streamlines may be obtained from the fact that in two dimensions, incompressibility via div w = implies that w = ( ψ y, ψ ) T, (3.9) x u=.5 u= u=.5.5.5 u= u=.5 y x Fig. 3.4. Contour plot (left) and three-dimensional surface plot (right) of an accurate finite element solution of Example 3..3, for ɛ = /2.

REFERENCE PROBLEMS 9 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.) 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.) 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..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( x 2 ), 2x( y 2 )) determines a recirculating flow with streamlines {(x, y) ( x 2 )( 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 =,y = ±. 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=.5 u= u=.5.5.5.5 u= y x Fig. 3.5. Contour plot (left) and three-dimensional surface plot (right) of an accurate finite element solution of Example 3..4, for ɛ = /2.