A CHARACTERISATION OF OSCILLATIONS IN THE DISCRETE TWO-DIMENSIONAL CONVECTION-DIFFUSION EQUATION

MATHEMATICS OF COMPUTATIO Volume 7, umber 4, Pages 63 88 S 5-578()39-8 Article electronically published on December 4, A CHARACTERISATIO OF OSCILLATIOS I THE DISCRETE TWO-DIMESIOAL COVECTIO-DIFFUSIO EQUATIO HOWARD C ELMA AD ALISO RAMAGE Abstract It is well known that discrete solutions to the convection-diffusion equation contain nonphysical oscillations when boundary layers are present but not resolved by the discretisation However, except for one-dimensional problems, there is little analysis of this phenomenon In this paper, we present an analysis of the two-dimensional problem with constant flow aligned with the grid, based on a Fourier decomposition of the discrete solution For Galerkin bilinear finite element discretisations, we derive closed form expressions for the Fourier coefficients, showing them to be weighted sums of certain functions which are oscillatory when the mesh Péclet number is large The oscillatory functions are determined as solutions to a set of three-term recurrences, and the weights are determined by the boundary conditions These expressions are then used to characterise the oscillations of the discrete solution in terms of the mesh Péclet number and boundary conditions of the problem Introduction Convection-diffusion equations arise in mathematical models of many different processes in diverse areas of science and engineering Analysis of the linear convection-diffusion equation () ɛ u(x, y)+w u(x, y) = f(x, y) in Ω u(x, y) = g(x, y) on δω, where the small parameter ɛ and divergence-free convective velocity field w = (w (x, y),w (x, y)) are given, is important in this context as it can be used to gain insight into the behaviour of these more complex phenomena as well as being of interest in its own right In this paper, we analyse the nature of the discrete solution produced when solving () using the Galerkin finite element method There are many textbooks which describe finite element modelling of convectiondiffusion equations in detail (see for example [], [4], [5]): we therefore present only a very brief outline here The weak form of () has solution u in the Sobolev space V = H(Ω) satisfying ɛ( u, v)+(w u, v) =(f,v) v V Received by the editor March 7, and, in revised form, February, Mathematics Subject Classification Primary 65, 653, 65Q5, 35J5 Key words and phrases Convection-diffusion equation, oscillations, Galerkin finite element method The work of the first author was supported by ational Science Foundation grant DMS99749 The work of the second author was supported by the Leverhulme Trust 63 c American Mathematical Society

64 HOWARD C ELMA AD ALISO RAMAGE Applying this to a finite-dimensional subspace V h of V, we seek u h V h such that () ɛ( u h, v)+(w u h,v)=(f h,v) v V h where f h is the L (Ω) orthogonal projection of f into V h OnagridwithM interior nodes, we may choose a set of basis functions Φ i, i =,,M for V h and look for an approximate solution of the form (3) u h (x, y) = M U i Φ i (x, y) i= Substituting this into () and choosing the test functions equal to the basis functions leads to a set of M linear equations (4) Au = f where the entries of u are the M unknown coefficients U i in (3) It is well known that applying the Galerkin finite element method to the onedimensional analogue of () will in certain circumstances result in a discrete solution which exhibits non-physical oscillations For linear elements on a uniform grid, a precise statement as to exactly when such oscillations occur can be made, namely, that for a problem with mesh size h, constant advective velocity w and different values at the left and right boundaries, oscillations will occur if the mesh Péclet number (5) P e = w h ɛ is greater than one (see for example [4], 3) The exact character of numerical approximations to the solution in this case is well understood The same is not true, however, for problems in two or more dimensions For example, Gresho and Sani ([], p 9) say of the two-dimensional case, useful, closed-form solutions are much harder to find and are often difficult to interpret even when found We know of very little analysis which has been done in this area, although Semper [6] presents a limited discussion of oscillations in streamline diffusion finite element solutions when ɛ = We are not aware of any work which describes the nature of oscillations in the full two-dimensional convection-diffusion equation In this paper we present a mechanism for deriving useful, closed-form twodimensional solutions to () and interpret the resulting formulae both in general and for a number of specific test problems In particular, we will characterise the behaviour of oscillations in the direction of the flow with respect to variations in the mesh Péclet number We begin in Section with a description of the basis of our analysis For the case where w =(, ) in () and the underlying finite element grid is uniform, we can use Fourier analysis to construct an analytic formula for the discrete solution u Specifically, we present a Fourier decomposition of A in (4) which means that blocks of the vector u can be explicitly expressed as a transformation of blocks of a vector y containing the solutions to a set of tridiagonal linear systems This idea is the basis of some fast direct solvers for linear systems of this type [8] Here, however, we take advantage of the fact that as these tridiagonal systems are of Toeplitz form, they can be solved analytically via a set of three-term recurrence relations: the construction and solution of these equations is described in Section 3 The result of this process is an exact analytic expression for the entries of the discrete solution vector u in (4) This takes the form of a

A CHARACTERISATIO OF OSCILLATIOS 65 weighted sum of three functions of the recurrence relation roots, where the weights depend on the boundary conditions of the problem In the remainder of the paper, we focus on the particular u arising from the Galerkin finite element method with bilinear basis functions on square elements ote, however, that the analysis is not specific to the finite element method but in fact applies to any discretisation method whose matrix entries fall within a certain nine-point stencil In Section 4, we obtain an analytic expression for the recurrence relation solution vector y in the bilinear case From this, we show that for a large mesh Péclet number, certain components of the recurrence relation solution are highly oscillatory and have behaviour analogous to solutions of simple onedimensional convection-diffusion problems In addition, we establish the fact that there is no mesh Péclet number for which the vector y is always guaranteed to be oscillation free Finally, we interpret these results in terms of the full discrete solution by examining the effects of the Fourier transform which relates the two vectors y and u By evaluating the boundary condition dependent weight functions, we can precisely characterise the behaviour of u for certain representative test problems Preliminary Fourier analysis The analysis presented in this section is based on Fourier techniques In order to use such an approach, we set w =(, ) and f= in () to obtain the equation () ɛ u + u y = inω=(, ) (, ), and we apply Dirichlet boundary conditions as shown in Figure for given functions f t, f r, f b and f l We will discretise this problem on a uniform grid of square elements with elements in each dimension (that is, grid parameter h =/ ) For many standard discretisation techniques with a natural ordering of the unknowns, the resulting linear system can be written in the form (4) where the (,) u=f t(x) (,) u=f (y) l u=f r(y) (,) u=f (x) b (,) Figure Boundary conditions

66 HOWARD C ELMA AD ALISO RAMAGE coefficient matrix A is of order ( ) and has the general structure M M M 3 M M A = () M 3 M M M 3 M Here, and M = tridiag(m,m,m ), M = tridiag(m 4,m 3,m 4 ), M 3 = tridiag(m 6,m 5,m 6 ) are all tridiagonal matrices of order Using discrete Fourier analysis, we can obtain analytic expressions for the eigenvalues and eigenvectors of the blocks of A: the matrices M, M and M 3 satisfy M v j = λ j v j λ j = m +m cos jπ (3) M v j = σ j v j σ j = m 3 +m 4 cos jπ M 3 v j = γ j v j γ j = m 5 +m 6 cos jπ for j =,,, where the eigenvectors are [ v j = sin jπ ] T, jπ ( )jπ (4) sin,,sin We now introduce a decomposition of the coefficient matrix A in term of these eigenvalues and eigenvectors First we consider the matrix V which has the vectors v j, j =,, as its columns and the related block diagonal matrix V which has V as each diagonal block We will also use diagonal matrices Λ = diag(λ i ), Γ=diag(γ i ) and Σ = diag(σ i ), i =,,, combining them to get V T AV = T = Λ Σ Γ Λ Σ Γ Λ Σ Γ Λ Introducing a permutation matrix P of order ( ),wemaywrite T T P T T P = T = T T where T i = tridiag(γ i,λ i,σ i ), that is, each block T i is a tridiagonal Toeplitz matrix of order Hence we have the decomposition A = VT V T = V(PTP T )V T

A CHARACTERISATIO OF OSCILLATIOS 67 Using this decomposition, (4) implies that is, V(PTP T )V T u = f P T V T u = T P T V T f, (5) u = VP y where the vector y is the solution to the linear system T y = P T V T f ˆf We recall that T is block diagonal: this system can therefore be partitioned into systems of the form (6) T i y i = ˆf i where T i is defined above and y and ˆf are partitioned in the obvious way ote that each vector y i contains the ith components of the one-dimensional Fourier transforms of the blocks of u Because T i is a Toeplitz matrix, each of these systems can be considered as a three-term recurrence relation which can be solved analytically to give an expression for each entry y ik of y i, k =,, As the rows of V are just the eigenvectors given in (4), the entry of the discrete solution vector (5) corresponding to the grid point (jh,kh) can now be written as (7) u jk = sin ijπ y ik i= 3 Solving the recurrence relations To see the exact form which the recurrences (6) take, we must consider more closely the structure of the right-hand side vector ˆf i As there is no forcing function in (), the only nonzero entries in the original right-hand side vector f arise from applying the Dirichlet boundary conditions In general, each entry in this vector consists of a sum of certain matrix coefficients times boundary values For example, suppose the + nodes on the bottom (or top) boundary are labelled from x to x ow construct ( )-vectors b and t with entries given by b i = m 6 f b (x i ) m 5 f b (x i ) m 6 f b (x i+ ), t i = m 4 f t (x i ) m 3 f t (x i ) m 4 f t (x i+ ) for i =,, (where the values m are the entries of A) Similarly, if the left (right) boundary nodes are labelled from y to y, construct vectors l and r with entries l k = m 6 f l (y k ) m f l (y k ) m 4 f l (y k+ ), r k = m 6 f r (y k ) m f r (y k ) m 4 f r (y k+ )

68 HOWARD C ELMA AD ALISO RAMAGE for k =,, With each of these, further associate ( ) -vectors ˆl = P T l b ˆb =, ˆt =, t l = l, ˆr = P T = r l r r r where l k =[l k,,,] T, r k =[,,,r k ] T and represents the zero ( )- vector The right-hand side vector f can now be written as f = ˆb + ˆt + ˆl + ˆr = b + s s s t + s where s k = l k + r k combines the left and right boundary condition contributions So V T (b + s ) b + s V T s s V T f = V T s s V T (t + s ) t + s and ˆf = P T V T f has blocks ˆf i with entries given by bi +( s ) i ( s ) i (3) ˆfi =, ( s ) i t i +( s ) i i =,, These vectors are precisely the right-hand side vectors for the systems in (6) In order to simplify the analysis presented in the remainder of the paper, we will henceforth assume that the functions f l (y) andf r (y) on the left and right boundaries are constant, that is, the vectors l k and r k (and hence s k ) are independent of k In this case, the vectors s k above can all be represented by a single vector s whose two nonzero entries are given by s = Lf l and s = Rf r,wheref l and f r are the (constant) boundary values and L = R = m 6 + m + m 4 Correspondingly, the vectors s k in (3) can all be replaced by the single transformed vector s,,

A CHARACTERISATIO OF OSCILLATIOS 69 Returning now to the systems (6), the solution of each is the solution of the equivalent three-term recurrence relation with constant coefficients γ i y i(k ) + λ i y ik + σ i y i(k+) = s i, y i = b i, y i = t i (3) γ i σ i (assuming γ i, σ i ) where the solution vector y i has entries y ik, k =,, Such an equation can be solved to obtain an explicit expression for y ik as follows The auxiliary equation is given by σ i µ + λ i µ + γ i = which has solutions µ (i) = λ i + λ i 4σ iγ i, µ (i) = λ i λ i 4σ i γ i (33) σ i σ i For notational convenience, we will frequently omit explicit reference to the i- dependence of these roots The complementary function is y ik = c µ k + c µ k for some constants c and c The right-hand side value s i is independent of k, so the constant particular function is given by and the general solution is y ik = s i σ i + λ i + γ i, y ik = c µ k + c µ k s i + σ i + λ i + γ i Applying the boundary conditions gives c = b i s i c, γ i σ i + λ i + γ i [ t i c = µ b ] i µ s i µ (µ ) σ i γ i σ i + λ i + γ i Hence the solution of the recurrence relation is y ik = t [ i µ k µ k ] σ i µ µ ( [ s i µ + ( µ k k ) ( µ ) µ k ]) We will write this as (34) σ i + λ i + γ i ( µ k µ b i γ i [ µ k µ k µ µ ]) y ik = F (i)g (i, k)+f (i)g (i, k)+f 3 (i)g 3 (i, k) = µ µ 3 F m (i)g m (i, k), m=

7 HOWARD C ELMA AD ALISO RAMAGE where, recalling the dependence of µ and µ on i, G (i, k) = µk µk µ, µ [ µ G (i, k) = ( µ k ) ( µ k ) µ k µ µ [ µ G 3 (i, k) = µ k µ k µ k ] µ, µ and the weight functions ], (35) F (i) = t i s i, F (i) =, σ i σ i + λ i + γ i F 3 (i) = b i γ i involve the coefficient matrix entries and boundary condition information ote that these weight functions are independent of k: forfixedi, the behaviour of y in the streamline (vertical) direction depends only on the functions G m (i, k) In addition, as G 3 (i, k) = G (i, k) G (i, k), we have the following result Theorem 3 The recurrence relation solution y ik has the form (36) y ik = F 3 (i)+[f (i) F 3 (i)] G (i, k)+[f (i) F 3 (i)] G (i, k) As F (i) is related to the top boundary values, F (i) is related to the sum of the left and right boundary values (which have been assumed to be constant for this analysis) and F 3 (i) is related to the bottom boundary values, this result shows that different boundary conditions will dictate how the functions G (i, k) andg (i, k) combine to produce different two-dimensional recurrence relation solutions y ik This point will be investigated further in Section 5 We will use either (34) or (36) to represent the entries of y, depending on which form is more useful at a particular point in the analysis Returning to (7), we see from (34) that u has entries (37) u jk = = 3 m= i= sin ijπ ( i= ( 3 ) F m (i)g m (i, k) m= sin ijπ F m(i)g m (i, k) for j, k =,, We emphasise that this is an explicit formula for the entries of the discrete solution vector u ) 4 Galerkin discretisation with bilinear elements The analysis above is applicable to many standard discretisation methods In this section, we will study the specific example of a Galerkin finite element discretisation of () using bilinear elements 4 The recurrence relation solution Theentriesofthecoefficientmatrix () in this case are m = 8 3 ɛ, m = 3 ɛ, m 3 = 3 [h ɛ], (4) m 4 = [h 4ɛ], m 5 = 3 [ h ɛ], m 6 = [ h 4ɛ]

A CHARACTERISATIO OF OSCILLATIOS 7 For convenience, we introduce the notation C i =cos iπ so that the eigenvalues (3) can be written as γ i = 6 [ ɛ( + C i) h( + C i )] (4) λ i = 3 [ɛ( + C i)+3ɛ( C i )] σ i = 6 [ ɛ( + C i)+h( + C i )] or γ i = 6 ( α α ), λ i = 3 (α + α 3 ), σ i = 6 ( α + α ), i =,,, where α = ɛ( + C i ), α = h( + C i ), α 3 =3ɛ( C i ) Substituting these into (33) gives the auxiliary equation roots µ, (i) = (α + α 3 ) ± 4(α α 3 + α 3 )+α α α To express these roots explicitly in terms of ɛ and h, we first examine the square root term, and write { 4(α α 3 + α 3)+α = h ( + C i ) + 3(5 + C } i)( C i ) ( + C i ), where P e = w h ɛ is the mesh Péclet number Also, { } (α + α 3 )=h (4 C i ) P e = ɛ, α α = h P e { ( + C i )+ } ( + C i ) P e Substituting these into µ, gives [ ] 4 Ci ± + 3(5 + C i)( C i ) +C i P e ( + C i ) Pe (43) µ, = [ ] +Ci +C i P e for the roots of recurrence relation (3) in the case of Galerkin approximation with bilinear elements

7 HOWARD C ELMA AD ALISO RAMAGE 4 When do oscillations occur? One question we would like to answer is, under what conditions is the recurrence relation solution (36) oscillatory? This point is addressed by the following theorem Theorem 4 If 3 <i, G (i, k) in (36) is an oscillatory function of k for any value of P e Proof We have (44) G (i, k) = µk µ k µ µ = ( µ µ ( µ ) k ) =Θ(i, k) µ k µk µ As µ /µ <, Θ(i, k) is always positive Hence if µ is negative, G (i, k)alternates in sign as k goes from to, that is, G (i, k) is oscillatory for fixed i From (43), the numerator of µ is always negative so we have the conditions P e <φ i µ >, G (i, k) is not oscillatory P e >φ i µ <, G (i, k) is oscillatory, where (45) φ i = +C i, +C i i =,, But P e > by definition and φ i < for 3 <i, so the second condition holds for these values of i independent of the value of P e Hence the corresponding functions G (i, k), 3 <i, are oscillatory for any value of P e Corollary 4 If the boundary conditions are such that the coefficient of G (i, k) in (36) is nonzero, the recurrence relation solution y will exhibit oscillations in the direction of the flow for any value of P e We conclude this section with a few brief remarks concerning the above theorem Remark Corollary 4 does not imply that the discrete solution u is oscillatory for any P e This issue will be explored in some detail in Section 5 Remark For other values of i, whether or not G (i, k) exhibits oscillations depends on the value of P e In particular, if P e >, then G (i, k) is an oscillatory function of k for every i {,, } Remark 3 From (44), the parity of the oscillations in G (i, k) is independent of i, that is, the sign of G (i, k) for a particular index k is the same for any i Remark 4 If P e = φ i for any i, then the eigenvalue σ i in (4) is zero and (3) reduces to a two-term recurrence relation This means that the analysis in Section 3 is not directly applicable, but the two-term recurrence can be solved in a similar way to obtain a formula for y ik : the details are omitted here Remark 5 As <µ <, G (i, k) =( µ k ) ( µ )G (i, k) will be oscillatory if and only if G (i, k) is oscillatory, although the oscillations will occur about the function µ k rather than zero in this case

A CHARACTERISATIO OF OSCILLATIOS 73 43 Characterising oscillations in the recurrence relation solution We now use (43) to gain a more detailed understanding of the functions G m (i, k), m =, in (36), with a view to characterising the oscillations which occur in the direction of the flow for large values of P e In particular, we highlight certain trends in the behaviour of these functions when ɛ h We begin by simplifying (43) using the approximation to the square root term given by { + 3(5 + C i)( C i ) ( + C i ) P e } / { + 3 } (5 + C i )( C i ) ( + C i ) Pe which assumes that Pe is small The formulae for µ and µ then become [ ] 4 Ci + 3 (5 + C i )( C i ) +C i P e ( + C i ) µ = P [ ] e, +Ci +C i P e [ ] 4 Ci 3 (5 + C i )( C i ) +C i P e ( + C i ) µ = P [ ] e +Ci +C i P e When these roots appear in the functions G and G in (36), they are raised to some power eglecting terms of order Pe and higher, we obtain the following approximate expressions for powers of the roots: ( [ ] ) k ( [ ] ) k 4 µ k Ci +Ci = +C i P e +C i P e ( [ ] )( [ ] ) 4 Ci k +Ci k + +C i P e +C i P e [ ] Ci k +3, +C i P e ( [ ] ) k ( [ ] ) k 4 µ k Ci +Ci = +C i P e +C i P e ( [ ] )( [ ] ) 4 ( ) k ( ) k Ci k +Ci k + +C i P e +C i P e [ ] ) ( ) (+ k 5+Ci k +C i P e ote that although the terms including Pe whichwehaveomittedheremayhave coefficients involving powers of (= /h), we are working under the assumption that ɛ h and so these terms are still small in size relative to those which we have retained We now use the results above to derive approximations to the functions G (i, k) and G (i, k) which appear in (36), simplifying the algebra by using the notation ( ) µ k k +ψ, µ k ( ) k k (46) +ψ P e P e

74 HOWARD C ELMA AD ALISO RAMAGE where [ ] Ci ψ (i) =3 and ψ (i) = 5+C i (47) +C i +C i With this notation, we have ( ) ( ) k G (i, k) = µk +ψ ( ) k k +ψ µk P e P µ ( ) ( e ) µ +ψ ( ) +ψ = P e P e [( ( ) k )+(ψ ( ) k ψ ) kpe ] ] [( ( ) )+(ψ ( ) ψ ) Pe We consider the cases of odd and even separately If is odd, we have ] G (i, k) [( ( ) k )+(ψ ( ) k ψ ) ][+(ψ kpe + ψ ) Pe Thus (48a) ][ [( ( ) k )+(ψ ( ) k ψ ) kpe + (ψ ] + ψ ) [ ] +Ci k + +C i P e G [ ] +Ci + +C i P e [ ] Ci 4 k +C i P e (k odd), G [ ] +Ci + +C i P e If is even, we have ][ G (i, k) [( ( ) k )+(ψ ( ) k ψ ) kpe (ψ ψ ) ] P e Thus (48b) G [ ] +Ci Pe C i 4 + [ ] +Ci k C i 4 (k odd), G k P e (k even) (k even) To characterise the behaviour of these functions for large P e, we consider the limit as P e (eg, as ɛ for a fixed value of ) In this limit, we have (49) G (i, k) is odd is even k is odd k k is even For both odd and even, G displays highly oscillatory behaviour in the streamline direction (that is, for fixed i) The nature of these oscillations is, however, slightly different: for odd, the oscillations remain bounded between and as P e, whereas for even, the oscillations grow unboundedly

A CHARACTERISATIO OF OSCILLATIOS 75 Similarly, we have G (i, k) =( µ k ) ( µ ) [ µ k µ k µ µ ] ψ P e (k G ) so if is odd, (4a) [ ] Ci (k ) 3 +C i P e G [ ] +Ci + +C i P e (k odd), G [ ] Ci k 3 +C i P [ ] e +Ci + +C i P e (k even), 5 8 6 4 5 4 5 6 8 3 4 5 6 7 8 9 5 3 4 5 6 7 8 9 (a) G : P e = (b) G : P e = 8 6 3 4 4 5 6 7 3 4 5 6 7 8 9 3 4 5 6 7 8 9 (c) G : P e = (d) G : P e = Figure 4 Plots of G and G (solid line, x) and approximations (dotted line, o) with =6andi =8

76 HOWARD C ELMA AD ALISO RAMAGE 5 5 5 5 5 5 3 4 5 6 7 8 9 5 3 4 5 6 7 8 9 (a) G : P e = (b) G : P e = 5 8 6 5 4 5 3 4 5 6 7 8 9 5 3 4 5 6 7 8 9 (c) G : P e = (d) G : P e = Figure 4 Plots of G and G (solid line, x) and approximations (dotted line, o) with =7andi =9 and if is even, [ ] [ ] ( + Ci )( 3C i ) k G (C i )( + C i ) + Ci (k odd), +C i (4b) [ ] Ci (k ) G 3 (k even) +C i P e In the limit as P e we have (4) G (i, k) is odd k is odd [ Ci +C i ] + is even k is even [ ] ( + Ci )( 3C i ) k (C i )( + C i )

A CHARACTERISATIO OF OSCILLATIOS 77 This time the two cases have different characters: if is odd, the oscillations in G (i, k) will die out as P e increases, whereas if is even, the oscillations remain, although their size is bounded independent of P e We point out here that the type of oscillations in the streamline direction which are caused by (48) and (4) are in each case of the same character as oscillations in certain discrete solutions of the one-dimensional convection-diffusion equation (4) ɛu + wu = f, u() = ξ, u() = ξ where w is a positive constant and f, ξ and ξ are given Specifically, consider the Galerkin method with linear elements applied to (4) with a uniform discretisation on + points in [, ] The function G (i, k) (forfixedi) exhibits similar asymptotic behaviour to the discrete solution of (4) with f =,ξ =,and ξ =,andg (i, k) (forfixedi) exhibits asymptotic behaviour like the discrete solution of (4) with f = /P e, ξ =andξ =(wherep e is the mesh Péclet number (5) of the one-dimensional problem) Representative plots of the exact (solid line, x) and approximate (dotted line, o) expressions for G (i, k) andg (i, k) forfixedi are plotted in Figures 4 and 4 ote that these plots have different scales: the oscillations in G are in general of greater magnitude than those in G The approximations (48) and (4) clearly capture the nature of the exact functions 44 The weight functions Recall from (36) that the functions F m (i), m =,, 3regulatehowG (i, k) andg (i, k) combine in y ik They therefore play an important role in dictating the nature of oscillations These weight functions come from transforming the right-hand side vector containing the boundary value and matrix coefficient information as described in Section 3 As shown in the Appendix, we can derive the expressions (43a) F (i) = p= f t (x p )sin piπ, F 3(i) = p= f b (x p )sin piπ, and, for the special case where the constant left and right boundary values f l and f r are equal, (43b) F (i) =f l p= sin piπ We may combine (43) and (36) to obtain the following result Theorem 4 If f l =f r, the recurrence relation solution y ik has the form y ik = f b (x p )sin piπ + [f t (x p ) f b (x p )] sin piπ G (i, k) p= p= (44) + [f l f b (x p )] sin piπ G (i, k) p= We emphasise the important difference between results (36) and (44): the latter expression involves the actual Dirichlet boundary functions themselves, as opposed to the more complicated weight functions F m (i) given by (35) The effect of each boundary condition on y ik is therefore much more readily seen in (44) If

78 HOWARD C ELMA AD ALISO RAMAGE there is a difference between the bottom and top boundary functions, this will introduce oscillations via G (i, k); similarly, a difference between the bottom boundary function and the left (right) boundary function will result in oscillations coming from G (i, k) 5 The full two-dimensional solution We have established that the recurrence relation solution y in (44) is influenced by two functions G and G which each look like the solution to a particular one-dimensional convection-diffusion problem In this section we explore the implications of this for the final two-dimensional solution u, that is, we examine the effect of the Fourier transformations (7) 5 The discrete solution u: An outflow boundary layer example Ideally, we would like to obtain an expression equivalent to (44) for u As (44) is a sum, we may examine the effect of transformation (7) term by term For the first term, this is straightforward: due to the orthogonality of the eigenvectors v j in (4), we have (5) i= = = p= p= sin ijπ = f b (x j ) f b (x p ) { { i= f b (x p )v T j v p p= } f b (x p )sin piπ jiπ sin piπ sin That is, the first term in u is just the bottom boundary function f b (x) Unfortunately, it is nontrivial to repeat this for the other terms in (44) due to the effect of the sine transform in (7) on each G m (i, k) We can, however, make a number of useful observations Firstly, there is the obvious point that if G or G has a zero coefficient in (44) due to equality in the relevant boundary functions, then the resulting u will also have no contribution from that function To illustrate a more subtle point, we look at the case where f t =andf b = f l = f r =,sothat u jk consists of a contribution from the function G alone Consider the solution u along a specified vertical grid line, that is, fix j From (7), (43) and (44) we have (5) u jk = = u jk = i= { i= { i= { } sin ijπ sin ipπ G (i, k) p= } sin ijπ sin ipπ G (i, k) p= } d ij G (i, k), }

A CHARACTERISATIO OF OSCILLATIOS 79 8 8 6 6 4 4 4 4 6 6 8 3 4 5 6 7 8 9 (a) G (,k) 8 3 4 5 6 7 8 9 (b) G (/,k) 8 6 4 4 6 8 3 4 5 6 7 8 9 (c) G (,k) Figure 5 Functions G (i, k) for =6andP e =5 where (53) d ij =sin ijπ p= sin ipπ That is, u jk is a linear combination of the functions G (i, k) fori =,, Examples of these functions for = 6 are plotted in Figure 5 for P e =5andin Figure 5 for P e =75 We can use the representation (5) to obtain insight into the quality of the solution The identity p= sin ipπ = sin iπ sin ( )iπ sin iπ

8 HOWARD C ELMA AD ALISO RAMAGE 5 5 5 5 5 5 5 5 3 4 5 6 7 8 9 (a) G (,k) 3 4 5 6 7 8 9 (b) G (/,k) 5 5 5 5 3 4 5 6 7 8 9 (c) G (,k) Figure 5 Functions G (i, k) for =6andP e =75 ([7], 94) leads to the simplified expression for the coefficients d ij, ( d ij = sin iπ )( sin ijπ ) ( ) cos iπ (54) sin iπ It follows immediately that d ij =forevenvaluesofi For the case j =,the nonzero values are d i =cos iπ, which are all positive ow recall from the proof of Theorem 4 that if P e >φ i in (45) then G (i, k) is an oscillatory function of k As we observed in subsection 4, if P e >, then G (i, k) is oscillatory for every i, with oscillations of the same parity in each case (see Figure 5) Thus, for j =,u jk is an oscillatory function of k, and we have established the following result:

A CHARACTERISATIO OF OSCILLATIOS 8 8 8 7 6 6 4 5 4 3 8 6 4 4 6 8 4 6 (a) j = 4 6 8 4 6 (b) j = /4 8 6 4 4 4 6 8 4 6 (c) j = / Figure 53 Coefficients d ij for = 6 Theorem 5 If P e > in the Galerkin discretisation of () with bilinear elements, then for some choice of boundary conditions the solution u exhibits oscillations This corresponds to the well-known restriction in the one-dimensional convectiondiffusion problem cited in the introduction In contrast to the one-dimensional case, the converse of Theorem 5 is not true in two dimensions To see this, let i be the lowest value of i {,,} such that P e >φ i and write i u jk = d ij G (i, k)+ d ij G (i, k) i= i=i = S smooth + S osc, where S smooth and S osc are sums of smooth and oscillatory functions respectively The overall behaviour of the solution u jk for a particular j is determined by the relative size of these two terms: if S smooth dominates, u jk will be smooth and if

8 HOWARD C ELMA AD ALISO RAMAGE 7 4 6 5 4 3 8 6 4 3 3 4 5 6 7 8 9 3 4 5 6 7 8 9 (a) j = : P e = 75 (b) j = 3: P e = 75 3 8 7 6 5 4 3 3 4 3 4 5 6 7 8 9 3 4 5 6 7 8 9 (c) j = : P e = 85 (a) j = 3: P e = 85 Figure 54 Comparison of S smooth (dotted line, o) and S osc (dashed line, o) with u jk (solid line, x) S osc dominates, u jk will be oscillatory As P e decreases, i increases so that most of the functions G (i, k) aresmoothands smooth contains most of the terms in (5) Furthermore, the magnitude of the nonzero coefficients d ij decreases rapidly as i goes from to : sample plots of d ij against i for =6areshowninFigure53 (note that we show values for j between and / onlyasd i( j) = d ij ) Thus, for small enough values of P e, S smooth will dominate In this sense, the Fourier transformations in (7) have a smoothing effect on the oscillatory recurrence relation solution y Figure 54 shows a comparison of S smooth (dotted line, o) and S osc (dashed line, o) with u jk (solid line, x) for = 6 with two values of P e which lead to different behaviour In plots (a) and (b), P e =75 and S smooth is larger in magnitude than S osc, producing a smooth u jk We have observed that the size of S osc decreases relative to that of S smooth as j goes from to /, so that S osc is most influential near the left and right boundaries (j =andj = ) For j as small as 3 (Figure 54(b)) it is already difficult to distinguish between the plots of S smooth and

A CHARACTERISATIO OF OSCILLATIOS 83 u 3k as S osc is very small Plots (c) and (d) show the equivalent data for P e =85 In this case, S osc is larger than S smooth when j =, resulting in an oscillatory u k The discrete solution u is therefore oscillatory, although P e < 5 Further examples and the effects of characteristic boundary layers To complete this discussion, we present some illustrations of the behaviour of u using a set of examples chosen to highlight some common features of convectiondiffusion problems Problem I Suppose we have the uniform boundary conditions f r (y) =f l (y) =f t (x) =f b (x) = which gives the exact solution u = everywhere This perfectly smooth solution can be recovered by setting f t = f b = f l = in (44) to get y ik = sin piπ ote that G and G are not present Under transformation (7) we obtain u jk =, as in (5) Problem II We return to the example considered in subsection 5 and set f t (x) =; f b (x) =f r (y) =f l (y) =, so that the solution has an exponential boundary layer of width ɛ along the top boundary As f b and f l are zero, the recurrence relation solution (44) is (55) y ik = sin piπ G (i, k), p= p= with resulting full solution (5) which is highly oscillatory for large P e Representative discrete solutions u for = 6 are illustrated in Figures 55 (a) and (d) The propagation of oscillations away from the boundary layer is caused by the behaviour of G This behaviour is typical of any problem where the top and bottom boundary conditions differ Problem III For this example, we choose f b (x) =f t (x) =; f l (y) =f r (y) =, so that the solution has characteristic layers on both sides of the domain In this case, recurrence relation solution (44) becomes y ik = p= sin piπ G (i, k) ote that this is the same as the recurrence relation solution (55) of Problem II, except with G in place of G The related full two-dimensional solution is { u jk = } (56) d ij G (i, k) i= (cf (5)) where the coefficients d ij are again given by (53) We may therefore apply the analysis of subsection 5 in the same way to Problem III, replacing G by G throughout In particular, u jk is a sum of smooth and oscillatory components

84 HOWARD C ELMA AD ALISO RAMAGE 8 6 4 4 6 8 5 5 5 5 3 3 5 8 8 6 8 6 8 4 4 6 4 4 6 (a) Problem II: P e = 5 (d) Problem II: P e = 5 8 8 6 6 4 4 8 8 6 8 6 8 4 4 6 4 4 6 (b) Problem III: P e = 5 (e) Problem III: P e = 5 5 5 5 5 8 8 6 8 6 8 4 4 6 4 4 6 (c) Problem IV: P e = 5 (f) Problem IV: P e = 5 Figure 55 Galerkin finite element solutions with = 6

A CHARACTERISATIO OF OSCILLATIOS 85 Moreover, G is oscillatory if and only if G is oscillatory (see Remark 5 at the end of subsection 4), so that the critical value of P e in terms of the onset of oscillations is the same for both problems The conclusion here is that there are oscillations in the streamline direction caused solely by the fact that the characteristic boundary conditions are different from those at the bottom (inflow) boundary; the precise effect of this difference can be seen in the last term of (44) Sample solutions for =6withP e =5andP e =5areshowninFigures 55 (b) and (e) It can be seen that the oscillations are much larger near the characteristic boundary layers than in the interior of the domain Some insight into this fact can be obtained from closer consideration of the components of (56) In particular, since the oscillations of G (i, k) with respect to k have the same parity, those of G will augment one another when their multipliers d ij have the same sign, but they will tend to cancel when the signs vary From (54), we see that for j =or (ie, for the grid lines next to the left and right boundaries) the coefficients d ij are all positive, whereas they alternate in sign for the other values of j shown in Figure 53, causing cancellation These trends are further accentuated by the fact that G (i, k) is larger for large i, but except for indices j near(and ), the corresponding multipliers d ij decrease in size dramatically as i increases Thus, the largest oscillations are near the layers, when j =and We have confirmed numerically, however, that the solutions for P e > oscillate along all vertical lines The pictures in Figure 55 also show that oscillations propagate away from the outflow boundary as P e increases We know from the analysis of G in subsection 43 that if is odd, these oscillations will die away as P e, although for any finite they are always present: for example, with P e =5, the largest oscillations are of magnitude 3 when = 7 as opposed to 5 when = 6 as in Figure 55 (e) Problem IV Here we apply the boundary conditions f b (x) =f t (x) =sin(πx); f r (y) =f l (y) = In the limit as ɛ, the solution of this problem is the smooth function u = sin(πx) everywhere As f b (x) =f t (x) andf l =, (44) gives y ik = sin pπ [ G (i, k)] sin piπ p= As in the previous example, there is no contribution from G (i, k) here, but there are oscillations in u for P e > due to the presence of G (i, k) Looking now at the full solution, we have { u jk = = i= i= sin ijπ [ G (i, k)] p= sin ijπ [ G (i, k)]v T v i = sin jπ [ G (,k)] } pπ piπ sin sin

86 HOWARD C ELMA AD ALISO RAMAGE The oscillations in this case are therefore relatively small as G (,k)issmallfor all k ote, however, that choosing f b (x) = f t (x) = sin(mπx) for an integer <m< increases the size of the oscillations: the solution would involve G (m, k) and, as noted above, G (i, k) is larger for large i In contrast to the previous example, the solution here is not dramatically worse next to the characteristic boundary layers; here the oscillatory part of the solution is almost independent of j 6 Summary In this paper, we have derived closed form expressions for discrete solutions to the two-dimensional convection-diffusion equation on a square that show how oscillations arise and are affected by boundary conditions Using bilinear finite elements as a concrete example, the analysis gives rigorous justification to the general belief that a mesh Péclet number greater than one leads to oscillatory solutions The analysis is applied to a problem with a unidirectional flow that is aligned with the grid However, even for this simple model, the results show several significant differences from the one-dimensional case In particular, it is possible for the discrete solution to contain oscillations even if the mesh Péclet number is less than one Moreover, the oscillations are affected by two-dimensional phenomena As with one-dimensional problems, oscillations may arise because Dirichlet boundary conditions are different at the inflow and outflow, but twodimensional effects, in particular, nearness to the side boundaries, also influence the size and quality of the oscillations: the solutions tend to be rougher near the side boundaries than in the middle of the domain In addition, oscillations may arise solely from characteristic boundary effects when the boundary conditions at the inflow differ from those along the sides, even if the inflow and outflow boundary conditions are the same Finally, we note that one way to reduce the extent of oscillations is to add artificial diffusion in the streamline direction, using the so-called streamline-diffusion method ([3], 97) The methodology introduced in this paper can be used to develop insight into this discretisation technique This topic is treated in the companion paper [] Appendix A The weight functions We derive formulae for the boundary condition dependent functions F m (i), i =,, 3 in (36) for a bilinear Galerkin finite element approximation We begin with F (i) = t i σ i Using the notation of Section 3, the vector t has entries t i = m 3 f t (x ) m 4 f t (x ), m 4 f t (x i ) m 3 f t (x i ) m 4 f t (x i+ ), i {,, } m 4 f t (x ) m 3 f t (x ) and so the ith entry of t = V T t is t i = t i,

A CHARACTERISATIO OF OSCILLATIOS 87 where t i =[ m 3 f t (x ) m 4 f t (x )] sin iπ + p= [ m 4 f t (x p ) m 3 f t (x p ) m 4 f t (x p+ )] sin piπ +[ m 4 f t (x ) m 3 f t (x )] sin = m 3 p= m 4 p= f t (x p )sin piπ m 4 f t (x p )sin piπ p= ( )iπ f t (x p+ )sin piπ { = f t (x p ) m 3 sin piπ m (p )iπ 4 sin m 4 sin p= { + f t (x ) m 3 sin iπ m 4 sin iπ } { } ( )iπ ( )iπ + f t (x ) m 3 sin m 4 sin = f t (x p )sin piπ { m 3 m 4 cos iπ } p= + f t (x )sin iπ { m 3 m 4 cos iπ } t i = ( )iπ + f t (x )sin p= Recalling from (3) that we have By a similar argument, Finally, consider f t (x p )sin piπ F (i) = { m 3 m 4 cos iπ { m 3 +m 4 cos iπ σ i = m 3 +m 4 cos iπ, t i σ i = F 3 (i) = b i γ i = F (i) = p= p= s i γ i + λ i + σ i } } f t (x p )sin piπ f b (x p )sin piπ } (p +)iπ

88 HOWARD C ELMA AD ALISO RAMAGE Assuming that the left and right boundary functions are constant, that is, f l (y k )=f l (y k )=f l (y k+ ) f l, f r (y k )=f r (y k )=f r (y k+ ) f r, we have s = (m + m 4 + m 6 )f l = ɛf l, s = (m + m 4 + m 6 )f r = ɛf r Hence the vector s = V T s has entries ( s i = ɛ f l sin iπ ) + f ( )iπ r sin = ɛ ( f l +( ) i+ ) iπ f r sin As γ i + λ i + σ i =(m + m 3 + m 5 )+(m + m 4 + m 6 )C i =ɛ( C i ), we obtain the expression F (i) = [ fl +( ) i+ ] iπ f r sin ( cos iπ ) For the special case where f l = f r, we have (see [7], 94) [ + ( ) i+ ]sin iπ F (i) =f l ( cos iπ ) = f l References p= sin piπ HC Elman and A Ramage, An analysis of smoothing effects of upwinding strategies for the convection-diffusion equation, Tech Report UMCP-CSD:CS-TR-46, University of Maryland, College Park MD 74, PM Gresho and RL Sani, Incompressible flow and the finite element method, John Wiley and Sons, Chichester, 999 3 C Johnson, umerical solutions of partial differential equations by the finite element method, Cambridge University Press, Cambridge, 987 MR 89b:653a 4 KW Morton, umerical solution of convection-diffusion problems, Chapman and Hall, London, 996 MR 98b:654 5 H-G Roos, M Stynes, and L Tobiska, umerical methods for singularly perturbed differential equations, Springer-Verlag, Berlin, 996 MR 99a:6534 6 B Semper, umerical crosswind smear in the streamline diffusion method, Comput Methods Appl Mech Engrg 3 (994), 99 8 MR 94k:766 7 MR Spiegel, Mathematical handbook of formulas and tables, Schaum s outline series, McGraw- Hill, ew York, 99 8 P Swarztrauber, The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson s equation on a rectangle, SIAMReview9 (977), 49 5 MR 55:639 Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, Maryland 74 E-mail address: elman@csumdedu Department of Mathematics, University of Strathclyde, Glasgow G XH, Scotland E-mail address: alison@mathsstrathacuk