MODIFIED STREAMLINE DIFFUSION SCHEMES FOR CONVECTION-DIFFUSION PROBLEMS. YIN-TZER SHIH AND HOWARD C. ELMAN y

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1 MODIFIED STREAMLINE DIFFUSION SCHEMES FOR CONVECTION-DIFFUSION PROBLEMS YIN-TZER SHIH AND HOWARD C. ELMAN y Abstract. We consider the design of robust and accurate nite element approximation methods for solving convection{diusion problems. We develop some two{parameter streamline diusion schemes with piecewise bilinear (or linear) trial functions and show that these schemes satisfy the necessary conditions for L -uniform convergence of order greater than = introduced by Stynes and Tobiska. For smooth problems, the schemes satisfy error bounds of the form O(h)juj in an energy norm. In addition, extensive numerical experiments show that they eectively reproduce boundary layers and internal layers caused by discontinuities on relatively coarse grids, without any requirements on alignment of ow and grid. Key words. Convection{diusion, streamline diusion, crosswind diusion, boundary layer, characteristic layer. AMS(MOS) subject classications. primary 65N3, 65F. Introduction. Consider the two{dimensional convection{diusion equation () () "u + ru = f in ; u = g where = ( ; ) is a ow velocity eld, " is a diusion or viscosity coecient, and f; g are given functions. For small values of ", it is well known that standard Galerkin nite element discretizations yield inaccurate oscillatory solutions near boundary layers, and if " is decreased without proportional reduction of the discretization mesh size h, then these inaccuracies propagate into regions where the solution is smooth [9], [, p. 59]. It is also known that these diculties can be ameliorated using the streamline diusion method [], a Petrov{Galerkin nite element method in which the test functions are modied to produce a small amount of articial diusion in the direction of streamlines, thereby enhancing stability. When solutions to (){() contain internal layers caused by discontinuities in boundary conditions, the approximate solution obtained by the streamline diusion method may suer from overshooting and undershooting along discontinuities [3], [4, p. 86]. \Shock{capturing" streamline diusion methods add a discontinuity capturing term (in the test functions) to reduce the local oscillations [3, 5]. Unfortunately, the shock{capturing term depends on the unknown discrete solution and this leads to a nonlinear discrete system even though the original problem is linear. An alternative approach is to add articial diusion in the crosswind direction, although this may lead to overly diuse numerical solutions in numerical experiments (see [6] and Section 5). In this paper, we present some new variants of the streamline diusion method that improve its performance. Our starting points are the papers of Roos [], Stynes and Interdisciplinary Applied Mathematics Program, University of Maryland, College Park, MD 74, yts@cs.umd.edu. y Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD 74, elman@cs.umd.edu. This work was supported by U. S. National Science Foundation under grant DMS

2 Tobiska [4], and Fischer, et. al. [8]. The rst two of these are concerned with necessary conditions on the structure of discretization operators for the convection{diusion equation for convergence of the discrete solution to be uniform in ". (Establishment of sucient conditions appears to be a dicult problem.) In particular, Roos derives three conditions needed for uniform convergence in L of order h, >, and Stynes and Tobiska show that two of Roos' conditions are needed for uniform convergence in L of order h, >. The third paper [8] considers the problem of choosing the parameter that determines the amount of articial diusion included in the streamline diusion discretization; it is shown that a good choice of this parameter leads to both small discretization errors and rapid convergence of certain iterative solvers for the discrete equations. Here, we build on these points by introducing some two{parameter versions of streamline diusion methods for bilinear (or linear) elements on uniform grids. The parameters determine the amount of articial diusion, but rather than being free parameters, they can be uniquely specied by requiring that the necessary conditions for L -uniform convergence established in [4] are satised. We present two versions of this idea that dier in their treatment of crosswind diusion. The contents of the rest of the paper are as follows. In Section, we outline some properties of the streamline diusion method and variants that add crosswind diusion. In Section 3, we summarize the necessary conditions for uniform convergence. In Section 4, we use the necessary conditions to derive the two new methods, and present an error analysis indicating that for linear or bilinear elements, they satisfy global error bounds in an energy norm of order hjuj. In Section 5, we present the results of numerical experiments on benchmark problems that demonstrate the eectiveness of the new methods.. Streamline diusion methods. For simplicity, we assume homogeneous Dirichlet boundary conditions on all boundaries; the ideas considered here generalize in a straightforward manner to other boundary conditions. Let V h = f v H () j v = g denote the nite element space. Let v = rv denote the derivative in the streamline direction. Let T = h denote a partitioning of. For our numerical examples, we will emphasize partitionings consisting of quadrilaterals; all the results of the paper apply naturally to triangles as well. The streamline diusion method (SD) is dened [4, p. 85] as: nd u h V h such that (3) B sd (u h ; v) = F v 8v V h ; where B sd (; ) is the bilinear form (4) (5) B sd (u h ; v) = B g (u h ; v) + s (u h ; v ) F v = (f; v) + s (f; v ); B g (u; v) = "(ru; rv) + (u ; v) denotes the bilinear form of the standard Galerkin method and (; ) denotes the usual scalar L inner product. On a uniform grid with mesh parameter h for which the mesh Peclet number kkh is greater than, the " parameter s is given by (6) s =! s h

3 where! s is a xed positive constant. The form B sd satises the coercivity condition B sd (v; v) "krvk L () + skv k L () 8v V h in which the lower bound is positive in the limit " &. In particular, the nite element matrix has positive denite symmetric part and the discrete solution u h of (3) is unique. If u is the strong solution and f L (), then Axelsson [] and Navert [8] have shown that for a piecewise linear nite element space there is a constant C (independent of h; s and ") such that " = kr(u u h )k + s = k(u u h ) k C " = h + = s where k k and j j denote the usual L norm and H seminorm respectively. It is also shown in [] that if s = O(h), " ch and r d > for positive constant d, then the error for SD satises h juj ; (7) ku u h k Ch 3= juj ; this is shown without a duality argument or elliptic regularity. In practice, when using SD to solve problems with characteristic internal and boundary layers, the choice of! s is crucial. Fischer et. al. [8] show that if % is the angle of ow to the horizontal, the choice (8)! s = kk " h j cos %j is a good choice with respect to both clustering of the spectrum of the discrete operator and performance of an iterative solution algorithm. Despite the success of SD, in practice, it suers from excessive overshooting and undershooting of front following characteristics when discontinuities are present [3]. Johnson, Schatz and Wahlbin [5] introduced a modication of the SD discretization that improves its performance by adding articial crosswind diusion. The streamline{ crosswind diusion method (SCD) as generalized by Lube [6] is given as follows: nd u Vh such that (9) B scd (u h ; v) = F v 8v V h ; where () B scd (u h ; v) = B sd (u h ; v) + (" m ")(u h ; v ); = ( ; ) is the crosswind vector and the articial crosswind diusion is dened by ( " for " h 3= " m = h 3= for " < h 3= : If T is not quasi{uniform or is a variable, then let h be the diameter of local element and determine s elementwise as in (6) (see [4, p. 86]). 3

4 For this method with piecewise linear elements, pointwise error bounds of order O(h j log hj) have been obtained for special meshes in [7], where it is also shown that the width of the characteristic boundary layers and interior layers along streamlines are of order h 5=8 log h. See also [, pp. 9.] for discussion of such results. Assume = (; ) (; ) and a square mesh is generated by the nodes (x i ; y j ) = (i h; j h), for i; j N, where N is a positive integer and the mesh size h = =N. The SD discretization (3) leads to a system of linear equations with (N ) unknowns. Following the notation in [8], the resulting coecient matrix can be expressed as () where A sd = "H + S + s U H i;j = (r j ; r i ); S i;j = ( r j ; i ); U i;j = ( r j ; r i ) and f i g (N ) i= are the nite element basis functions. The right hand side b is given by b i = (f; i + s r i ): If r =, then for each basis function i having value on the boundary, it follows from integration by parts that ( r j ; i ) = ( r i ; j ); that is, S is skew-symmetric. It is then easy to see that the symmetric part, "H + s U, is positive denite. For constant and bilinear basis functions, the constituent 9-point stencils are as follows: () (3) (4) (5) where H : 3 S : U : 6 6 h h ( + ) The coecient matrix of the SCD discretization can be expressed as A scd = "H + S + s U + (" m ")C C i;j = ( r j ; r i ): The matrix "H + s U + (" m ")C is also symmetric and positive denite, and the 9{point stencil of C is given by (6) C : ( + ) In our numerical experiments, we nd that this method dramatically reduces the oscillations of discrete solutions near boundary layers and internal layers, although there are still problems with smearing near sharp fronts (see Section 5) : 3 7 5

5 3. Necessary conditions. In this section, we outline some convergence criteria and analysis for the dependence of discrete solutions of (){() on the diusion coecient ". In particular, let h be a set of triangulations of depending on a mesh parameter h. We will say that convergence of the discrete solution u h = u ";h is uniform in " if, for varying ", and the parameter h chosen such that h=" is constant, there is an error bound of the form (7) ju u h j Ch ; where >, C is independent of ", and j j is a norm. Convergence criteria of this type are considered in [], [9], [] and [4]. Here, we are concerned with necessary conditions for (7) to hold for the L norm and >. Assume and are constant and the discretization is on a uniform grid using bilinear or linear elements. The discrete problem then has the form (8) X ;= ;; a ; u h i+;j+ = h ^f i;j ; for i; j N; where ^f depends only on f, u h i;j = uh (x i ; y j ) and P ;= ;; a ; =. Theorem 3.. [4] Let f L () such that the solution u o of the reduced problem (9) ( ruo = f in u o = is in C ( )\C () = f(x; j n < ; n is an outward unit normal vectorg. Assume that the scheme (8) yields a solution u h that satises ku u h k Ch uniformly in " for >. Then the coecients of the scheme must satisfy! X X X! a ; e h " + a ; + a ; e () h " = and () X! X a ; e h " + a ; + X! a ; e h " = : For the streamline diusion method, the \stencil" coecients fa ; g depend linearly on the parameter! s. In the case = =, the two conditions () and () are identical and they are satised by the choice ()! s = coth h " " h If the streamlines are aligned with the grid (e.g. = ) then one of the equations (() in this case) is degenerate and the choice! s = coth h " " (3) h satises (). These results coincide with a well known \optimal value" in the one{ dimensional problem (see [6]) for which the discrete solution is exact at the nodes. Otherwise, if 6=, then () and () constitute two linear equations with a single unknown! s, for which there is no solution [4]. Therefore, the streamline diusion method does not satisfy these necessary conditions for uniform convergence. 5! :

6 4. Two{parameter streamline diusion schemes. Using the observations of the previous section as motivation, we now present two variants of the streamline diusion method that contain two free parameters, which allow the necessary conditions of (){() to be satised. One of these techniques (the rst) was presented for linear elements by Roos [, p. 465]. In this section, we will assume that the velocity eld satises r d for constant d. A case of primary interest is where both inequalities here are replaced by equality, as in the incompressible Navier-Stokes equations. We will also consider the case d > ; this is a technical assumption [] which is used in the derivation of L -error bounds. (An alternative is to include a zero-order term in (), see [4], [5], [6].) Experimental results described in Section 5 use benchmark problems in which is divergence free. In our rst variant, we add crosswind diusion to the weak formulation, producing the new weak form (denoted SD-A) (4) where (5) and c. 3 B sda (u h ; v) = F v 8v V h ; B sda (u h ; v) = B sd (u h ; v) + c (u h ; v ) For r d, let jv j = "krvk + s kv k + c kv k + d kvk : The stability of method (4) is a consequence of the following result. Lemma 4.. The bilinear form B sda satises (6) B sda (v; v) jv j Proof. By Green's formula, for any v V h (): = and thus (v ; v) = v n ds = R Z r (v ) d = r v d d kvk. Hence Z r v d + (v ; v); B sda (v; v) "krvk + s kv k + c kv k + d kvk : Therefore, the SD-A discretization has a unique solution. We have the following error estimate, the proof of which follows [], [5], [6]. Theorem 4.. Let u be the solution of (){() with g = and u H () \ H (). Let W ; () and either r = or r d >, for constant d. If u h is the discrete solution obtained by SD-A on either bilinear or linear elements, then the discretization error satises (7) ju u h j C " = h + s = h + = for constant C >. c h + s = h + h + c juj ; There it was observed that this scheme does not satisfy the conditions for L {uniform convergence when 6=. 3 If T is not quasi{uniform or is a variable, then let h be the diameter of local element and determine s and c elementwise as in SD. 6

7 Proof. Let = u I u, where u I is the bilinear (or linear) interpolant of u. It follows that (8) (9) kk ch juj ; krk chjuj j j c " = h + s = h + c = h + h juj (see [4, p. 76], [, p. 3]). Setting = u I u h yields (3) j j B sda (; ) = B sda (; ) + B sda (u u h ; ): The quasi{orthogonality relation holds, where P er(u; v) is the truncation error B sda (u u h ; v) = P er(u; v) for all v V h ; P er(u; v) = "(u; s v ) + c (u ; v ): But Green's formula and the Poincare inequality leads to B sda (u u h ; ) "s = kuks = k k + c (k(u ) k + kr k ku k) kk "s = juj + c c juj j j (3) "s = + c c juj j j and Z B sda (; ) = "(r; r) + s ( ; ) + c ( ; ) (; ) (r ) d: If r = (divergence free case), then B sda (; ) "krk krk + s k k k k + c k k k k + kk k k " = krk + s = k k + c = k k + s = kk j j: If r d >, then Z (r ) d ckk kk c kk j j and B sda (; ) " = krk + s = k k + c = k k + s = kk + c kk j j: Use (8){(9) to get (3) B sda (; ) c " = h + s = h + c = h + s = h + h juj j j: Combining (3), (3) and (3), gives j j c " = h + s = h + c = h + s = h + h + "s = + c juj : 7

8 Thus, using the triangle inequality, it follows that ju u h j j j + j j C " = h + s = h + c = h + s = h + h + "s = + c juj : As shown in [5], if " ch, then for the choices s = O(h) and c = O(h 3= ), all the terms of (7) depending on s and c are of same order of magnitude, and the error bound (7) implies that ju u h j Ch 3= juj. When r d >, one can also get an L bound ku u h k Ch 3= juj which is the same estimate for SD shown in []. We now consider a dierent approach for choosing s and c based on [4]. Suppose the discretization is on a uniform square mesh of width h with bilinear shape functions. The resulting coecient matrix has the form (33) A sda = "H + S + s U + c C; where C is as in (6). For 6= ; 6=, the presence of two parameters in (33) allows us to impose conditions (){(). The result is a system of linear equations (34) (35) s + c = h coth h " "; s + c = h coth h " ": (The case = or = is addressed in Section 3.) For 6=, this system is nonsingular so the parameters s ; c are uniquely determined and have the values (36) (37) s = c = h 3 coth h " 3 coth h " jj " h coth h " coth h " jj " In case of =, we may choose the limit values of s and c as!. Let us compare these choices with s dened by (6), (8) and c = h 3=. Assume " h and = cos 6= ; = sin 6=. The function coth is very close to for large, and therefore the quantities in parentheses are essentially constant. Consequently, s of (36) and c of (37) are eectively proportional to h. This implies in particular that the asymptotic bound of (7) is of size O(h), larger than that of SCD. Figure expands on this point. Plot (a) on the left compares the values of s =h from (36) to! s from (8) and indicates that SD-A includes more diusion in the upwind direction. Plot (b) compares c =h to h = for several values of h; for moderate values of h or values of near o or 9 o, the amount of added crosswind diusion is comparable to that of SCD. In the second two{parameter variant of SD, we introduce an extra parameter by modifying the contribution to the discrete diusion operator in the coordinate dimension in which the ow is weaker. That is, the weak form (SD-B) is given by! ;! : (38) B sdb (u h ; v) = F v 8v V h ; 8

9 h=/.54. delta_s / h delta_c / h.5..5 h=/4 h=/8 h=/ theta theta (a) Solid lines are s=h from (36); dashdot line is! s from (8). (b) Solid lines are c=h from (37); dashdot lines are h =. Fig.. The values of s=h; c=h for " = 5 ; = (cos ; sin ); o 9 o and h = =; =4; =8; =6. where (39) B sdb (u h ; v) = B sd (u h ; v) + ~"(u h ; v ); = ( y if < j j < j j x if < j j < j j; and ~". If j j = j j or the streamlines are aligned with the grid, then we use ~" = and impose the necessary conditions using the SD method as in Section 3. The stability and error bounds for the SD-B scheme are summarized below. The proofs are analogous to those for SD-A and we omit them. Lemma 4.3. If r d for constant d, then (4) B sdb (v; v) "krvk + s kv k + ~"kv k + d kvk for any v H (): Theorem 4.4. Let u h be the discrete solution obtained by SD-B on either bilinear or linear elements. Under the assumptions of Theorem 4., we have ju u h j C " = h + s = h + ~" = h + s = h + h + ~" juj : As above, the parameters s and ~" can be used to satisfy the conditions (){(). The coecient matrix derived from (39) is A sdb = "H + S + s U + ~"P; P i;j i 9

10 Consider the case when the streamlines are not aligned with the mesh, for example, > >. Then the necessary conditions (){() lead to " h s e h " + " + s + " + h s e h " = ~" " h s e h " + ~" + " + s + ~" " + h s e h " = : That is, (4) (4) s = h coth h " "; ~" = h coth h " s ": delta_s / h epsilon_tilde / h theta (a) Solid lines are s=h from (4); dashdot line is! s from (8) theta (b) ~"=h obtained by (4) Fig.. The values of c and ~" for " = 5 ; = (cos ; sin ); o 45 o and h = =; =4; =8; =6. Let us compare these choices with s dened by (6) and (8). In Figure, for " = 5 ; = (cos ; sin ); o 45 o and various mesh sizes h = =; =4; =8; =6, Plot (a) on the left compares the value of s =h from (4) to! s from (8) and indicates that SD-B also includes more diusion in the upwind direction. Plot (b) shows the values of ~"=h obtained by (4) which determines the amount of crosswind-like diusion. We see that ~" dissipates when the streamlines are aligned with the mesh or =. Remark 4: We are using the necessary conditions for the uniform convergence as a means of specifying the parameters in these discretizations. The conditions are not known to be sucient for uniform convergence and we are not considering this issue here. Cf. [, pp. 73.] for other discretizations that display uniform convergence. 5. Numerical experiments. The analytic results used and cited in Section 4 are for smooth problems, i.e., u H (). However, the real need for streamline diffusion methods occurs for problems with steep boundary layers or discontinuities. In this section, we present the results of numerical experiments that show the performance of the new streamline diusion methods for such problems and compare it to

11 that of the classical methods SD and SCD. We use three test problems. One has a known solution given by the sum of two one{dimensional examples and containing downstream boundary layers, one has a downstream boundary layer and a characteristic internal layer, and one has variable ow. All experiments use bilinear shape functions on square elements on a uniform N N element grid with h = =N, and they were performed with MATLAB Version 4.c on a SUN SPARC{ workstation. The coecient s =! s h of the streamline diusion term for both SD and SCD was chosen using (8) (a) SD (b) SCD (c) SD-A (d) SD-B Fig. 3. Numerical solutions for problem for " = 4 ; = 75 o ; h = =. Table Maximum absolute errors at the nodes for h = =. " SD SCD SD-A SD-B :e 8:58e 3 8:55e 3 :3e 5 :e 5 :e 4 4:3e 5 :8e 5 4:3e 5 4:3e 5 5 o :e :8e :55e 3:9e 6 :e 6 :e 4 5:97e 3:e 9:5e 7 :7e 6 45 o :e :94e :68e 6:9e 7 9:7e 7 :e 4 3:89e :e 3:3e 6 :e 46 Problem : Downstream boundary layers/analytic solution. We rst consider the problem with exact known solution given by u(x; y) = e x=" e =" + ey=" e =" ; where ( ; ) = (cos ; sin ) for o < < 9 o. (For = o, u(x; y) = ex=" + y, the e =" limit value.) The Dirichlet boundary conditions and right hand side f are determined

12 from the exact solution. Figure 3 depicts the numerical solutions for " = 4 ; = 75 o and h = =. The results indicate that the two new schemes reduce the oscillations or excess diusivity exhibited by SD and SCD, respectively. Indeed, for these problems we nd that the new schemes reproduce the exact solution at the nodes essentially to within machine precision, as shown in Table. Problem : Characteristic and downstream boundary layers. This problem was rst considered in [] for studying a downstream boundary layer and a characteristic internal layer that propagates along the characteristics when inow boundary conditions are discontinuous. Assume the velocity eld is given by (cos ; sin ), and boundary values are as follows: ( if y < =; x = or y = ; x < u = otherwise. The reduced problem (9) has discontinuous solution ( y < u = x + y > x + : For " >, there is an internal layer of width O( p ") across the characteristic y = x +, and a boundary layer of width O(") at x = [7] y.5 x y.5 x..5 (a) SD (b) SCD y.5 x y.5 x..5 (c) SD-A (d) SD-B Fig. 4. Numerical solutions and contours for " = 5 ; h = = and = 5 o. We rst examine the accuracy with which the internal layer is approximated by the various discretizations. Figure 4 depicts the three{dimensional structure and contour plots of the numerical solutions obtained by the four methods tested, for " = 4 ; h = = and = (cos 5 o ; sin 5 o ). Figure 5 shows contour plots for " = 5 and several mesh sizes. It is evident that the solutions obtained by SD oscillate around the internal layer and those obtained by SCD are overly diuse. The two new variants appear to

13 SD SCD SD-A SD-B (a) 8 8 grids (b) 6 6 grids (c) 3 3 grids Fig. 5. Contour plots showing numerical internal and boundary layers for " = 5 and = 5 o SCD : o SD A : + SD B : x crosswind smear width / h Fig. 6. The crosswind smear width (y) when mesh is rened and " = 5 ; = 3 and = 5 o Table The measure of overshoots and undershoots for " = 5. h shooting SD SCD SD-A SD-B =6 u + u :e 3:44e 5:49e 5 :e 4 :79e 4 =3 u + :e 9:54e 7 :3e 5 u 4:6e 6:e 6 :9e 5 =64 u + 9:78e 3:6e 3:7e 9 :7e 5 u 3:44e 6:8e 4 5:45e 8 3:5e 7 3

14 produce a more accurate representation of the internal layer. We can quantify this statement as follows. Let n o y j u h (:5; y) y u = min y n o ; y l = max y j u h (:5; y) ; for small > : y Then y = y u y l is a measure of the width of the numerical internal layer and the eect of crosswind smearing at x = :5. Figure 6 plots this quantity for various mesh sizes and = 3. The results indicate that the two{parameter schemes introduce less crosswind smearing than SCD. Moreover, the lower-order O(h) asymptotic bound (Theorem 4:) obtained for smooth problems does not appear to have a deleterious eect as h & in this nonsmooth example. It is known that SD produces overshooting and undershooting about sharp layers [3]. We can use a similar device to examine the size of these overshoots and undershoots. Let u + = max y n o n o u h (:5; y) ; u = min u h (:5; y) : y Thus u + ; u give a measure of the sizes of overshooting and undershooting respectively, at x = :5. Table shows the eectiveness of SD-A and SD-B in this measure. Remark 5: These results for = (cos 5 o ; sin 5 o ) are representative of our experience with other velocity elds = (cos ; sin ) for ( 9 o ; 9 o ). Problem 3: Variable ow eld. For variable ows, we can dene local (to element) values of parameters determining the amounts of articial diusion. We follow the approach given in [3]: on any element, let (x ; y ) denote the element center, let = (x ; y ), and let these constant values be used to dene the parameters in formulas (8), (36){(37) and (4){(4) in the local matrix computations associated with the element. Note that a similar approach could also be used for irregular grids. For our third benchmark problem, with variable ow, we consider two variants of the \IAHR/CEGB" workshop problem [3] in common use for testing discretization strategies (see e.g. [], [7]). As in [3], let the domain be the rectangular region and the velocity eld be = f(x; y) j < x < ; < y < g ; = (y( x ); x( y )): The inow boundary is the interval f(x; ) j x < g, and Dirichlet conditions specied there represent an inlet temperature which is convected in a circular ow to the outow boundary f(x; ) j < x g, where natural boundary = ; for < x are assigned. For both our variants, Dirichlet boundary conditions are given on the remainder The rst choice is dened by values (43) u(x; ) = + tanh( + x) 4

15 y.5.5 x..5 y.5.5 x (a) SD (b) SCD y.5.5 x..5 y.5.5 x (c) SD-A (d) SD-B Fig. 7. Numerical solutions and contours for Problem 3, rst variant with " = 5 and h = = y.5 x y.5.5 x (a) SD (b) SCD y.5 x y.5 x (c) SD-A (d) SD-B Fig. 8. Numerical solutions and contours for Problem 3, second variant with " = 5 and h = =6. 5

16 at the inlet together with u = at x = and y =. This is essentially the problem of [3] and it has also been examined in []. It contains a fairly smooth characteristic internal layer and no boundary layer. Representative pictures of the three-dimensional structure and contour plots of the numerical solutions obtained by the four methods tested, for " = 5 and h = =6; are shown in Figure 7. (The three-dimensional plots are rotated 8 o to give a clearer picture of the layer.) Our second variant of this problem uses the value u = (a hot wall) at x = as in [7]; this introduces a thin boundary layer at the right boundary. In addition, we add a discontinuity in the inlet prole (44) u(x; ) = ( x < :5 :5 x : The homogeneous Dirichlet conditions at x = and y = remain intact. The results for this example, again for " = 5, h = =6, are shown in Figure 8. Consideration of Figure 7 shows that for the (relatively smooth) example determined by (43) and the homogeneous condition at x =, the best solution is obtained by pure streamline-diusion (SD). The new variants SD-A and SD-B produce solutions that are somewhat overdiuse, but quite a bit less so than SCD. We also remark that the solution obtained by the Galerkin method without any upwinding is qualitatively similar to that obtained by SD. Figure 8 shows that the situation is dierent when discontinuities are present. None of the methods eliminate oscillations completely (showing that none are monotonic), but the new methods clearly improve the accuracy of the solution near the boundary layer. They also produce more accurate approximations to the internal layer than SCD, and, in contrast with those obtained by SD, these components of the solutions are nonoscillatory. There may still be some smearing of the crosswind layer; here SD-B seems to be slightly more eective than SD-A. REFERENCES [] O. Axelsson. On the numerical solution of convection dominated convection-diusion problems. In K. I. Gross, editor, Mathematics Methods in Energy Research, pages 3{. SIAM, Philadelphia, 984. [] O. Axelsson, V. Eijkhout, B. Polman, and P. Vassilevski. Incomplete block-matrix factorization iterative methods for convection-diusion problems. BIT, 9:867{889, 989. [3] A. Brooks and T. Hughes. Streamline upwind/petrov-galerkin formulations for convection dominated ows with particular emphasis on the incompressible Navier-Stokes equations. Comp. Meth. Appl. Mech. Engng., 3:99{59, 98. [4] C. Johnson and U. Navert. An analysis of some nite element methods for advection-diusion problems. In O. Alexsson, L.S. Frank, and A. van der Sluis, editors, Analytical and Numerical Approaches to Asymptotic Problem in Analysis. North-Holland, Amsterdam, 98. [5] C. Johnson and U. Navert and J. Pitkaranta. Finite element methods for linear hyperbolic problems. Comp. Meths. Appl. Mech. Engrg., 45:85{3, 984. [6] I. Christie, D.F. Griths, A.R. Mitchell, and O.C. Zienkiewicz. Finite element methods for second order dierential equations with signicant rst derivatives. Int. J. Numer. Meth. Engrg., :389{396, 976. [7] W. Eckhaus. Boundary layers in linear elliptic singular perturbation problems. SIAM Review, 4:5{7, 97. [8] B. Fischer, A. Ramage, D. Silvester, and A.J. Wathen. Towards parameter-free streamline upwinding for advection-diusion problems. Technical Report 37, Department of Mathematics, University of Strathclyde,

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