c 2004 Society for Industrial and Applied Mathematics

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1 SIAM J. NUMER. ANAL. Vol. 4, No. 3, pp. 6 9 c 004 Society for Industrial and Applied Mathematics CONVERGENCE ANALYSIS OF A MULTIGRID METHOD FOR A CONVECTION-DOMINATED MODEL PROBLEM MAXIM A. OLSHANSKII AND ARNOLD REUSKEN Abstract. The paper presents a convergence analysis of a multigrid solver for a system of linear algebraic equations resulting from the discretization of a convection-diffusion problem using a finite element method. We consider piecewise linear finite elements in combination with a streamline diffusion stabilization. We analyze a multigrid method that is based on canonical intergrid transfer operators, a direct discretization approach for the coarse-grid operators and a smoother of line- Jacobi type. A robust diffusion and h-independent bound for the contraction number of the two-grid method and the multigrid W-cycle are proved for a special class of convection-diffusion problems, namely with Neumann conditions on the outflow boundary, Dirichlet conditions on the rest of the boundary, and a flow direction that is constant and aligned with gridlines. Our convergence analysis is based on modified smoothing and approximation properties. The arithmetic complexity of one multigrid iteration is optimal up to a logarithmic term. AMS subject classifications. 65F0, 65N, 65N30, 65N55 Key words. multigrid, streamline diffusion, convection-diffusion DOI. 0.37/S Introduction. Concerning the theoretical analysis of multigrid methods, different fields of application have to be distinguished. For linear self-adjoint elliptic boundary value problems the convergence theory is well developed cf. [5, 9, 35, 36]. In other areas the state of the art is far less advanced. For example, for convectiondiffusion problems the development of a multigrid convergence analysis is still in its infancy. In this paper we present a convergence analysis of a multilevel method for a special class of two-dimensional convection-diffusion problems. An interesting class of problems for the analysis of multigrid convergence is given by { u + b u = f in Ω=0,,. u = g on Ω, with >0and b = cos φ, sin φ, φ [0, π. The application of a discretization method results in a large sparse linear system which depends on a mesh size parameter h. For a discussion of discretization methods for this problem we refer to [8,, ] and the references therein. Note that in the discrete problem we have three interesting parameters: h mesh size, convection-diffusion ratio, and φ flow direction. For the approximate solution of this type of problems robust multigrid methods have been developed which are efficient solvers for a large range of relevant values for the parameters h,, φ. To obtain good robustness properties the components in the multigrid method have to be chosen in a special way because, in general, the Received by the editors November 6, 00; accepted for publication in revised form October 8, 003; published electronically October 8, Department of Mechanics and Mathematics, Moscow M. V. Lomonosov University, Moscow 9899, Russia Maxim.Olshansii@mtu-net.ru. The research of this author was partially supported by Russian Foundation for Basic Research grants and lined to project Institut für Geometrie und Pratische Mathemati, RWTH Aachen, Templergraben 55, D-5056 Aachen, Germany reusen@igpm.rwth-aachen.de. 6

2 6 MAXIM A. OLSHANSKII AND ARNOLD REUSKEN standard multigrid approach used for a diffusion problem does not yield satisfactory results when applied to a convection-dominated problem. To improve robustness several modifications have been proposed in the literature, such as robust smoothers, matrix-dependent prolongations, and restrictions and semicoarsening techniques. For an explanation of these methods we refer to [9, 33, 4, 3, 4, 8, 9, 37]. These modifications are based on heuristic arguments and empirical studies and rigorous convergence analysis proving robustness is still missing for most of these modifications. Related to the theoretical analysis of multigrid applied to convection-diffusion problems we note the following. In the literature one finds convergence analyses of multigrid methods for nonsymmetric elliptic boundary value problems which are based on perturbation arguments [6, 9, 7, 3]. If these analyses are applied to the problem in. the constants in the estimates depend on and the results are not satisfactory for the case. In [, 5] multigrid convergence for a one-dimensional convection-diffusion problem is analyzed. These analyses, however, are restricted to the one-dimensional case. In [3, 6] convection-diffusion equations as in. with periodic boundary conditions are considered. A Fourier analysis is applied to analyze the convergence of two- or multigrid methods. In [3] the problem. with periodic boundary conditions and φ = 0 is studied. For the discretization the streamline diffusion finite element method on a uniform grid is used. A bound for the contraction number of a multigrid V-cycle with point Jacobi smoother is proved which is uniform in and h provided h is satisfied. Note that due to the fact that a point Jacobi smoother is used one can not expect robustness of this method for h 0. In [6] a two-grid method for solving a first order upwinding finite difference discretization of the problem. with periodic boundary conditions is analyzed, and it is proved that the two-grid contraction number is bounded by a constant smaller than one which does not depend on any of the parameters, h,φ. In [3] the application of the hierarchical basis multigrid method to a finite element discretization of problems as in. is studied. The analysis there shows how the convergence rate depends on and on the flow direction, but the estimates are not uniform with respect to the mesh size parameter h. In [7] the convergence of a multigrid method applied to a standard finite difference discretization of the problem. with φ = 0 is analyzed. This method is based on semicoarsening and a matrix-dependent prolongation and restriction. It is proved that the multigrid W-cycle has a contraction number smaller than one independent of h and. The analysis in [7] is based on linear algebra arguments only and is not applicable in a finite element setting. Moreover, the case with standard coarsening, which will be treated in the present paper, is not covered by the analysis in [7]. In the present paper we consider the convection-diffusion problem. u + u x = f in Ω:=0,, u x = 0 on Γ E := { x, y Ω x =}, u =0 on Ω \ Γ E. In this problem we have Neumann boundary conditions on the outflow boundary and Dirichlet boundary conditions on the remaining part of the boundary. Hence, the solution may have parabolic layers but exponential boundary layers at the outflow boundary do not occur. For this case an a priori regularity estimate of the form u H c f L holds, whereas for the case with an exponential boundary layer one only has u H c 3 f L. Due to the Dirichlet boundary conditions a Fourier analysis is not applicable.

3 MULTIGRID FOR A CONVECTION-DOMINATED PROBLEM 63 For the discretization we use conforming linear finite elements. As far as we now there is no multigrid convergence analysis for convection-dominated problems nown in the literature that can be applied in a finite element setting with nonperiodic boundary conditions and yields robustness for the parameter range 0 h. In this paper we present an analysis which partly fills this gap. We use the streamline diffusion finite element method SDFEM. The SDFEM ensures a higher order of accuracy than a first order upwind finite difference method cf. [8, 38]. In SDFEM a mesh-dependent anisotropic diffusion, which acts only in the streamline direction, is added to the discrete problem. Such anisotropy is important for the high order of convergence of this method and also plays a crucial role in our convergence analysis of the multigrid method. In this paper we only treat the case of a uniform triangulation which is taen such that the streamlines are aligned with gridlines. Whether our analysis can be generalized to the situation of an unstructured triangulation is an open question. We briefly discuss the different components of the multigrid solver. For the prolongation and restriction we use the canonical intergrid transfer operators that are induced by the nesting of the finite element spaces. The hierarchy of coarse grid discretization operators is constructed by applying the SDFEM on each grid level. Note that due to the level-dependent stabilization term we have level-dependent bilinear forms and the Galerin property A = r A p does not hold. Related to the smoother we note the following. First we emphasize that due to a certain crosswind smearing effect in the finite element discretization the x-line Jacobi or Gauss Seidel methods do not yield robust smoothers i.e., they do not result in a direct solver in the limit case = 0; cf. [9]. This is explained in more detail in Remar 6. in section 6. In the present paper we use a smoother of x-line-jacobi type. These components are combined in a standard W-cycle algorithm. The convergence analysis of the multigrid method is based on the framewor of the smoothing- and approximation property as introduced by Hacbusch [9, 0]. However, the splitting of the two-grid iteration matrix that we use in our analysis is not the standard one. This splitting is given in 6.8. It turns out to be essential to eep the preconditioner corresponding to the smoother W in 6.8 as part of the approximation property. Moreover, in the analysis we have to distinguish between residuals which after presmoothing are zero close to the inflow boundary and those that are nonzero. This is done by using a cut-off operator Φ in 6.8. The main reason for this distinction is the following. As is usually done in the analysis of the approximation property we use finite element error bounds combined with regularity results. In the derivation of a L bound for the finite element discretization error we use a duality argument. However, the formal dual problem has poor regularity properties, since the inflow boundary of the original problem is the outflow boundary of the dual problem. Thus Dirichlet outflow boundary conditions would appear and we obtain poor estimates due to the poor regularity. To avoid this, we consider a dual problem with Neumann outflow and Dirichlet inflow conditions. To be able to deal with the inconsistency caused by these wrong boundary conditions we assume the input residuals for the coarse grid correction to be zero near the inflow boundary. Numerical experiments from section related to the approximation property show that such analysis is sharp. In our estimates there are terms that grow logarithmically if the mesh size parameter h tends to zero. To compensate this the number of presmoothings has to

4 64 MAXIM A. OLSHANSKII AND ARNOLD REUSKEN be taen level dependent. This then results in a two-grid method with a contraction number T A T A c< and a complexity ON ln N 4, with N = h. Using standard arguments we obtain a similar convergence result for the multigrid W-cycle. The remainder of this paper is organized as follows. In section we give the wea formulation of the problem. and describe the SDFEM. In section 3 some useful properties of the stiffness matrix are derived. In section 4 we prove some a priori estimates for the continuous and the discrete solution. In section 5 we derive quantitative results concerning the upstream influence of a right-hand side on the solution. These results are needed in the proof of the modified approximation property. Section 6 contains the main results of this paper. In this section we describe the multigrid algorithm and present the convergence analysis. In sections 7 0 we give proofs of some important results that are used in the analysis in section 6. In section we present results of a few numerical experiments.. The continuous problem and its discretization. For the wea formulation of the problem. we use the L Ω scalar product which is denoted by,. For the corresponding norm we use the notation. With the Sobolev space V := { v H Ω v =0 on Ω \ Γ E } the wea formulation is as follows: find u V such that. au, v :=u x,v x +u y,v y +u x,v=f,v for all v V. From the Lax Milgram lemma it follows that a unique solution of this problem exists. For the discretization we use linear finite elements on a uniform triangulation. For this we use a mesh size h := and grid points x i,j =ih,jh, 0 i, j h. A uniform triangulation is obtained by inserting diagonals that are oriented from southwest to northeast. Let V V be the space of continuous functions that are piecewise linear on this triangulation and have zero values on Ω \ Γ E. For the discretization of. we consider the SDFEM: find u V satisfying. + δ h u x,v x +u y,v y +u x,v=f,v + δ h v x for all v V with.3 δ = { δ if h, 0 otherwise. The stabilization parameter δ is a given constant of order. For an analysis of the SDFEM we refer to [8, 5]. In this paper we assume [ ].4 δ 3,. The value 3 for the lower bound is important for our analysis. The choice of for the upper bound is made for technical reasons and this value is rather arbitrary. The finite element formulation. gives rise to the stabilized bilinear form.5 a u, v := + δ h u x,v x +u y,v y +u x,v, u,v V. Note the following relation for the bilinear form a, : a v, v = v y + + δ h v x +.6 v dy for v V. Γ E

5 MULTIGRID FOR A CONVECTION-DOMINATED PROBLEM 65 The main topic of this paper is a convergence analysis of a multigrid solver for the algebraic system of equations that corresponds to.. In this convergence analysis the particular form of the right-hand side in., which is essential for consistency in the SDFEM, does not play a role. Therefore for an arbitrary f L Ω we will consider the problems.7 u V such that a u, v =f,v for all v V,.8 u V such that a u,v =f,v for all v V. Note that u and u depend on the stabilization term in the bilinear form and that these solutions differ from those in. and.. 3. Representation of the stiffness matrix. We now derive a representation of the stiffness matrix corresponding to the bilinear form a, that will be used in the analysis below. The standard nodal basis in V is denoted by {φ l } l N with N the dimension of the finite element space, N := h isomorphism: N P : X := R N V, P x = x i φ i. i= h. Define the On X we use a scaled Euclidean scalar product x, y = h N i= x iy i and corresponding norm denoted by note that this notation is also used to denote the L Ω norm. The adjoint P : V X satisfies P x, v = x, P v for all x X,v V. The following norm equivalence holds: 3. C x P x C x for all x X, with a constant C independent of. The stiffness matrix A on level is defined by 3. A x, y = a P x, P y for all x, y X. In an interior grid point the discrete problem has the stencil 3.3 h h , := + δ h. For a matrix representation of the discrete operator we first introduce some notation and auxiliary matrices. Let n := h and ˆD x := tridiag,, 0 R n n, h  x := ˆD x T ˆD x = h  y := h tridiag,, R n n, Rn n, Ĵ :=... R n n, ˆT := tridiag0, 0, R n n.

6 66 MAXIM A. OLSHANSKII AND ARNOLD REUSKEN Furthermore, let I m be the m m identity matrix. We finally introduce the following N N matrices D x := I n ˆD x, A x := I n Âx = D T x D x, A y := Ây Ĵ and the N N bloctridiagonal matrix B := bloctridiagi n, 4I n, ˆT. Using all this notation we consider the following representation for the stiffness matrix A in 3.: 3.4 A = + δ h A x + A y BD x. The latter decomposition can be written in stencil notation as h h h with = +δ 3 h > 0. Some properties of the matrices used in the decomposition 3.4 are collected in the following lemma. For B,C R n n we write B C iff x T Bx x T Cx for all x R n. Lemma 3.. The following inequalities hold: A x Dx 0, A y Dx 0, B I, A D x 3 I, D x A 3. Proof. To chec 3.6 observe A x Dx = Dx T D x Dx = Dx T. Now note that Dx T + D x is symmetric positive definite. To prove 3.7 it suffices to show that Dx T A y 0 holds. We have K := D T x A y =I n ˆD T x Ây Ĵ =Ây D T x, with the matrix D x T = h Hence in the matrix K + K T = Ây D T x + D x both factors Ây and D T x + D x are symmetric positive definite. From this the result follows.

7 MULTIGRID FOR A CONVECTION-DOMINATED PROBLEM 67 To prove 3.8 we define R := B 4I and note that R R R 4. Using this we get Bx,x =4 x + Rx, x 4 x R x x which proves the desired result. Inequality 3.9 follows immediately from the representation of A in 3.4 and inequalities From the result in 3.9 it follows that D T x A 3 DT x D x. This implies D x x 3 A x, D x x 3 A x D x x for all x X and thus estimate 3.0 is also proved. 4. A priori estimates. In this paper we study the convergence of a multigrid method for solving the system of equations 4. A x = b, with A the stiffness matrix from the previous section. As already noted in the introduction, our analysis relies on smoothing and approximation properties. For establishing a suitable approximation property we will use regularity results and a priori estimates for solutions of the continuous and the discrete problems. Such results are collected in this section. In the remainder of the paper we restrict ourselves to the convection-dominated case. Assumption 4.. We consider only values of and such that h. If instead of the factor in this assumption we tae another constant C, our analysis can still be applied but some technical modifications are needed to distinguish between δ = δ and δ = 0 which mae the presentation less transparent. We consider this convection-dominated case to be the most interesting one. Many results that will be presented also hold for the case of an arbitrary positive but the proofs for the diffusion-dominated case often differ from those for the convectiondominated case. In view of the presentation we decided to treat only the convectiondominated case. Note that then 4. δ = δ [ ] 3, and 3 h = + δh 3 h. For the inflow boundary we use the notation Γ W := {x, y Ω x =0}. For the continuous solution u the following a priori estimates hold. Theorem 4.. For f L Ω let u be the solution of.7. There is a constant c independent of and such that u + u x c f, uy c f, h u xx + h u xy + u yy c f, u dy + h Γ E u x dy + Γ W u y dy c f. Γ E Proof. Since f L Ω, the regularity theory from [8] ensures that the solution u of.7 belongs to H Ω. Hence we can consider the strong formulation of.7, 4.7 u yy u xx + u x = f,

8 68 MAXIM A. OLSHANSKII AND ARNOLD REUSKEN with boundary conditions as in.. Now we multiply 4.7 with u x and integrate by parts. Taing boundary conditions into account, we get the following terms: u yy,u x = u y x, = u y dy, Γ E u xx,u x = u x x, = u x dy ch u x dy we use 4., Γ W Γ W u x,u x = u x u, f,u x f + u x. From these relations the results 4.3 and 4.6, except the bound for Γ E u dy, easily follow. Next we multiply 4.7 with u and integrate by parts to obtain u y + u x + Γ E u dy =f,u f u c f we use 4.3. Estimate 4.4 and the remainder of 4.6 now follow. To prove 4.5 we introduce F = f u x. Due to 4.3 we have F c f. Moreover u yy u xx = F holds. If we square both sides of this equality and integrate over Ω we obtain 4.8 u yy + u yy,u xx + u xx = F c f. Further note that for any sufficiently smooth function v, satisfying the boundary conditions in., the relations v xx x, 0 = v xx x, =0,x 0,, v y 0,y=v xy,y=0,y 0,, hold, and thus v yy,v xx = v y,v xxy =v xy,v xy. Using a standard density argument we conclude that for the solution u H Ω of.7 the relation u yy,u xx =u xy,u xy holds. Now 4.8 gives u yy + u xy + u xx c f. In combination with 4. this yields 4.5. The next lemma states that the x-derivative of the discrete solution is also uniformly bounded if the right-hand side is from V. Lemma 4.. For f V let u V be a solution to.8; then 4.9 u x c f. Proof. The result in 4.9 follows from the estimate 3.0 in Lemma 3.. To show this we need some technical considerations. First we show how the size of the x-derivative of a finite element function v V can be determined from its corresponding coefficient vector P v X. Let I be the index set {i, j 0 i n, j n }. For i, j Ilet Ti,j l and be the two triangles in the triangulation which have the line between the grid T u i,j

9 MULTIGRID FOR A CONVECTION-DOMINATED PROBLEM 69 points x i,j and x i+,j as a common edge. Let v V be given. For j n we introduce the vector v j =vx,j,...,vx n,j T. We then obtain v x = vx dxdy + vx dxdy i,j I T l T u i,j i,j = vxi+,j vx i,j h = h Dx v j T D x v j Therefore 4.0 = h i,j I Dx P h v T Dx P v = D x P v. j n v x = D x P v for any v V. For the discrete solution of.8 with f = f we have the representation u = P A P f. Now from 3.0 and 4.0 it follows that u x = D x A P f 3 P f c f with a constant c independent of and. The next lemma gives some bounds on the difference between discrete and continuous solutions Lemma 4.3. Define the error e = u u, where u and u are solutions of the problems.7 and.8 with right-hand side f = f V. Then the following estimates hold: e y + e x c f Γ E e dy c h f. Proof. Estimate 4. directly follows from 4.3 and 4.9 by a triangle inequality. The proof of 4. is based on standard arguments: the Galerin orthogonality, approximation properties of V, and a priori estimates from 4.5. Indeed e y + + δh e x + e dy = a e,e = inf a e,u v Γ v V E inf e y u v y + + δh e x u v x + e x u v v V c h e y u H + h e x u H c h e y f + h f e y + c h f. The estimate 4. follows. 5. Upstream influence of the streamline diffusion method. Consider the continuous problem.7. The goal of this section is to estimate the upstream influence of the right-hand side function f on the solution u. The same will be done for the corresponding discrete problem. In the literature, results of such type are nown for the problem with Dirichlet boundary conditions and typically formulated

10 70 MAXIM A. OLSHANSKII AND ARNOLD REUSKEN in the form of estimates on the discrete Greens function see, e.g., [3, 0, 6]. A typical result is that the value of the solution at a point x is essentially determined by the values of the right-hand side in a small strip that contains x. This strip has a crosswind width of size O ln h, where = max{, h 3 }, and in the streamline direction it ranges from the inflow boundary to a Oh ln h upstream distance from x. In our analysis we need precise quantitative results for the case with Neumann outflow boundary conditions. In the literature we did not find such results. Hence we present proofs of the results that are needed for the multigrid convergence analysis further on. Our analysis uses the nown technique of cut-off functions e.g., [7, 6], it avoids the use of an adjoint problem and is based on the following lemma. Lemma 5.. For = + δh assume a function φ H 0,, such that 0 φ x φ. Denote by φ a semi-norm induced by the scalar product φ,. Then the solution u of.7 satisfies 5. u x φ f φ, 5. φ0 u x dy f φ, Γ W u φ x + u y φ φf,u. Proof. We consider the strong formulation 4.7 and multiply it with φu x and integrate by parts. We then get the following terms: u yy,φu x = u y φ x + Γ φ u y dy 0, E u xx,φu x = u x φ x + φ0 u x dy Γ W u x φ + φ0 u x dy, Γ W u x,φu x = u x φ, f,φu x f φ u x φ f φ + 4 u x φ. Now 5. and 5. immediately follow. To obtain the estimate 5.3 we multiply 4.7 with φu and integrate by parts. We get the following terms: u yy,φu= u y φ, u xx,φu= u x φ + u x,φ x u u x φ u x φ x 4 u φ x 4 u φ x, u x,φu= u φ x + φ u dy. Γ E Thus 5.3 follows. For arbitrary ξ [0, ] consider the function { for x [0,ξ], φ ξ x = exp x ξ for x ξ,]. For any ξ the function φ ξ x satisfies the assumptions of Lemma 5.. For 0 <ξ< η< we define the domains Ω ξ = {x, y Ω: x<ξ}, Ω η = {x, y Ω: x>η}.

11 MULTIGRID FOR A CONVECTION-DOMINATED PROBLEM 7 Direct application of Lemma 5. with φ = φ ξ gives the following corollary. Corollary 5.. Consider f L Ω such that suppf Ω η and let u be the corresponding solution of problem.7. Assume η ξ p ln h, p > 0. Then we have u x LΩ ξ h p f, u x dy h p f, Γ W uy LΩ ξ h p f. Proof. The estimate f φ =φf, f Ω η φη f Ω η = h p f and 5., 5. imply the results 5.4 and 5.5. We also have φf, u =φf, u Ωη f φ + φu, u Ωη = f φ φ xu, u Ωη f φ + 4 u φ x. Together with 5.3 this yields 5.6. We need an analogue of estimate 5. for the finite element solution u of.8. To this end consider a vector φ =φ 0,...,φ n, such that φ i > 0 for all i and φ i φ i h c 0 φ i, i =,...,n, with a constant c 0 0, 4 9 and = + δh. Define ˆΦ := diagφ i i n,φ := I n ˆΦ with φ i satisfying 5.7. Let, Φ = Φ,. Lemma 5.3. There exists a constant c>0 independent of and such that A x, D x x Φ c D x x Φ for all x X. Proof. We use similar arguments as in the proof of 3.0. We use the representation 3.4 of the stiffness matrix: A = A x + A y + 6 BD x. Note that D T x Φ A y =I n ˆD T x I n ˆΦ Ây Ĵ =Ây ˆD T x ˆΦ Ĵ. The matrix Ây is symmetric positive definite. Using φ i φ i and a Gershgorin theorem it follows that ˆD x T ˆΦ Ĵ + Ĵ ˆΦ ˆDx is symmetric positive definite, too. Hence, Dx T Φ A y 0 holds, i.e., 5.8 A y x, D x x Φ 0 for all x X. From the assumption on φ it follows that φ i + c0h φ i for all i. Using this and the relation ˆΦ ˆD T ˆΦ x + ˆΦ ˆD x ˆΦ = φ i φ i tridiag,, h φ i it follows that φ i+

12 7 MAXIM A. OLSHANSKII AND ARNOLD REUSKEN holds. And thus Φ Dx T Φ + c 0h I c 0 I c 0 I h 5.9 A x x, D x x Φ = Φ D T x D x x, D x x c 0 D x x, D x x Φ for all x X. We decompose B as B =4I R. A simple computation yields Φ RΦ + + c 0h + +3c c 0. Similarly we get Φ RΦ + 3 c 0 and thus Φ RΦ + 3 c 0. Hence Φ BΦ c 0 I = 3 c 0 I and thus BD xx, D x x Φ 3 4 c 0 D x x, D x x Φ for all x X. Combination of the results in 5.8, 5.9, and 5.0 yields A x, D x x Φ c 0 D x x, D x x Φ c D x x, D x x Φ for all x X with a constant c>0 use that c 0 0, 4 9. Lemma 5.4. For f = f V let u be the solution of the problem.8. Then 5. n i= n j= h ui,j u i,j φ i C h n i= n j= h φ i M ˆf i,j holds. Here u ij is the nodal value of u at the grid point x i,j, ˆf is the vector of nodal values of f, M is the mass matrix, and φ i satisfies 5.7. Proof. Let û = P u X be the vector of nodal values of u ; then 5. A û = M ˆf =: ˆb. The diagonal matrices Φ and ˆΦ are as in Lemma 5.3. The statement of the lemma is equivalent to Φ D x û,d x û c Φ ˆb, ˆb, with a constant c that is independent of ˆb. This is the same as 5.3 D x A Φ c. Note that 5.3 is a generalization of the result in 3.0. From Lemma 5.3 we obtain D x x ˆΦ < c A x, D x x ˆΦ c A x Φ D x x Φ for all x X ; thus D x x Φ c A x Φ for all x. Hence we have proved the result in 5.3.

13 MULTIGRID FOR A CONVECTION-DOMINATED PROBLEM 73 For the discrete case we consider 5.4 { φ ξ i = exp ih ξ 4h for ih [0,ξ], for ih >ξ. It is straightforward to chec that φ ξ i φξ i = exp 4 φξ i if ih > ξ. Therefore, using 3 h, 5.5 φ ξ i 0 φξ i 3 exp φ ξ i, i =,,.... h 4 For any ξ the vector φ ξ i, i n, satisfies the condition 5.7 with c 0 = 3 exp 4. This constant is less than 4 9. As a consequence of Lemma 5.4 we obtain discrete versions of the results in Corollary 5.. Corollary 5.5. Consider f V such that suppf Ω η and let u be a the corresponding solution of problem.8. Assume η ξ 8 h p ln h, p > 0; then u x LΩ ξ ch p f, u y LΩ ξ cξh p f. Proof. Estimate 5.6 is a consequence of 5.. Indeed, observe the following inequalities: u x LΩ ξ c i: ih ξ = c c i: ih ξ n j= n j= h max φ n i ih η ui,j u i,j h h ui,j u i,j φ i c h i= n j= h M h ˆf i,j c n i= n j= h φ i M ˆf i,j max φ i f ch p f. ih η Estimate 5.7 follows from an inverse inequality, the Friedrichs inequality, and 5.6: u y LΩ ξ ch u LΩ ξ cξh u x LΩ ξ cξh p f. Corollary 5.6. Consider f V such that suppf Ω η. Let u and u be the solutions.7 and.8, respectively. Assume η ξ 8 h p ln h,p>0. Then for e = u u we have e x LΩ ξ ch p f, { e y LΩ ξ c max ; } ξ h p f. h Proof. The proof is made by direct superposition of estimates in Corollaries 5. and 5.5. The result in Corollary 5.6 shows that the H -norm of errors close to the inflow boundary can be made arbitrarily small if the right-hand side is zero on a sufficiently

14 74 MAXIM A. OLSHANSKII AND ARNOLD REUSKEN large subdomain Ω \ Ω η that is adjacent to this inflow boundary. In the proof of the approximation property in section 0 we will need these estimates for the case ξ = h and p =. Hence we tae η =4h ln h + h. Note that for the results in the previous corollaries to be applicable we need right-hand side functions f which are zero in Ω \ Ω η. For technical reasons we tae Ω η such that the right boundary of the domain Ω \ Ω η coincides with a grid line. We use ln h = ln and thus 4h ln h + h 3 +h and introduce the following auxiliary domains for each grid level: 5.8 Ω in := { x, y Ω x<3 +h }. As a direct consequence of the previous corollary we then obtain the following. Corollary 5.7. Consider f V such that f is zero on the subdomain Ω in. Let u and u be the solutions of.7 and.8, respectively. Then for e = u u we have e x LΩ h ch f, e y LΩ h c h f. 6. Multigrid method and convergence analysis. In this section we describe the multigrid method for solving a problem of the form A x = ˆb with the stiffness matrix A from section and present a convergence analysis. For the prolongation and restriction in the multigrid algorithm we use the canonical choice: 6. p :X X, p = P P, r = 4 pt. Let W :X X be a nonsingular matrix. We consider a smoother of the form 6. x new = S x old, ˆb =x old ω W A x old ˆb for x old, ˆb X, with corresponding iteration matrix denoted by 6.3 S = I ω W A. The preconditioner W we use is of line-jacobi type: 6.4 W = 4 h I + D x. Note that W is a blocdiagonal matrix with diagonal blocs that are n n bidiagonal matrices. A suitable choice for the parameter ω follows from the analysis below. Remar 6.. In the literature it is often recommended to apply a so-called robust smoother for solving singularly perturbed elliptic problem using multigrid. Such a smoother should have the property that it becomes a direct solver if the singular perturbation parameter tends to zero cf. [9], chapter 0. In the formulation 6. one then must have a splitting such that A W = O the constant in O may depend on. Such robust smoothers are well nown for some anisotropic problems. For anisotropic problems in which the anisotropy is aligned with the gridlines one

15 MULTIGRID FOR A CONVECTION-DOMINATED PROBLEM 75 can use a line Jacobi or Gauss Seidel method or an ILU factorization as a robust smoother. Theoretical analyses of these methods can be found in [9, 30, 34]. If the convection-diffusion problem. is discretized using standard finite differences it is easy to see that an appropriate line solver yields a robust smoother. However, in the finite element setting such line methods do not yield a robust smoother. This is clear from the stencil in 3.3. For 0 the diffusion part yields an x-line difference operator which can be represented exactly by an x-line smoother, but in 6 0] parts of the difference operator the convection stencil the [0 6 6 ] and [ 6 are not captured by such a smoother. It is not clear to us how for the finite element discretization, with a stencil as in 3.3, a robust smoother can be constructed. In multigrid analyses for reaction-diffusion or anisotropic diffusion problems one usually observes a dependence in the standard approximation property that is then compensated by an factor from the smoothing property cf. [,, 9, 30, 34]. However, we cannot apply a similar technique, due to the fact that for our problem class a robust smoother is not available. Instead, we use another splitting of the iteration matrix of the two-grid method, leading to modified -independent smoothing and approximation properties. We consider a standard multigrid method with pre- and postsmoothers of the form as in 6., 6.4. In the analysis we will need different damping parameters for the pre- and postsmoother. Thus we introduce S,pr := I ω,pr W A, S,po := I ω,po W A. We also define the transformed iteration matrices S,pr := A S,pr A, S,po := A S,po A. We will analyze a standard two-grid method with iteration matrix 6.5 T = S ν,po I p A r A S µ,pr. For the corresponding multigrid W-cycle the iteration matrix cf. [0] is given by 6.6 M mgm 0 := 0, M mgm = T + S ν,po p M mgm A r A S µ,pr, >. In the convergence analysis of this method the auxiliary inflow domain Ω in defined in 5.8 plays a crucial role. As in the analysis of the upstream influence in section 5 we will use a cut-off function in the x-direction. We define diagonal matrices ˆΦ,Φ as follows: 6.7 ξ := 3 +h, ˆΦ := diagφ ξ,...,φξ n, Φ := I n ˆΦ ; here φ ξ i is the cut-off function defined in 5.4 with ξ =3 +h. For notational simplicity we drop the superscript ξ in φ ξ i in the remainder. Note that the diagonal matrix Φ is positive definite. For any symmetric positive definite matrix C R m m we define x, y C := x T Cy, x C := x, x C, B C := C BC with x, y R m, B R m m. Note that if C = E T E for some nonsingular matrix E then B C = EBE.

16 76 MAXIM A. OLSHANSKII AND ARNOLD REUSKEN The convergence analysis is based on the following splitting, with A := A : 6.8 T A T A = S ν,po I p A r A S µ,pr A T A = S ν,po A p A r I Φ +Φ A S µ,pr A T A S ν,po A p A r I Φ A S µ,pr A T A + S ν,po A p A r Φ A S µ,pr A T A S ν,po A W W A + S ν,po I A p A r Φ Sµ,pr. p A r I Φ S µ,pr Remar 6.. Note that the splitting in 6.8 differs from the usual splitting that is used in the theory based on the smoothing and approximation property introduced by Hacbusch cf. [0]. In this theory the approximation property of the form A p A r C A gh, is combined with a smoothing property of the form A S µ,po ηµ gh, with some ηµ such that ηµ 0, µ uniformly with respect to h and. In numerical experiments we observed that bounds of this type are not liely to be valid. Due to the fact that the smoother is not an exact solver for 0 cf. Remar 6., it is essential to have the preconditioner W as part of the approximation property. Furthermore, it turns out that for obtaining an appropriate bound for W A p A r f the right-hand side function f must vanish near the inflow boundary. We illustrate this by numerical experiments in section. This motivates the introduction of the cut-off matrix Φ in the decomposition. We now formulate the main results on which the convergence analysis will be based. The proofs of these results will be given further on. Theorem 6.. The following holds: 6.9 W A I for =,, Proof. The proof is given in section 7. Lemma 6.. From 6.9 it follows that I ωa W [0, for all ω ]. 4 Proof. The proof is elementary. Assumption 6.. In the postsmoother S,po we tae ω,po := 8. We note that the analysis below applies for any fixed ω,po 0, 8 ]. We obtain the following smoothing property. Corollary 6.. There exists a constant c independent of and such that 6.0 S ν,po A W c ν. Proof. Follows from Lemma 6. and Theorem in [0] or results in [, 4]. The result holds with c = 3 π. We now turn to the presmoother.

17 MULTIGRID FOR A CONVECTION-DOMINATED PROBLEM 77 Theorem 6.3. There exist constants d > 0,d > 0 independent of and such that Φ I d 6. A W Φ d 4. Proof. The proof is given in section 8. Assumption 6.. In the presmoother S,pr we tae ω,pr := min{ 4, d }. Remar 6.3. The result in 6. can be written as I d A W Φ d. 4 Hence, we have a contraction result in the almost degenerated norm Φ. This norm, however, coincides with the Euclidean one for the vectors that have a support only in Ω in. Hence the result in 6. indicates that the presmoother is a fast solver near the inflow boundary cf. section. Concerning the approximation property the following result holds. Theorem 6.4. There exists a constant c independent of and such that 6. W A p A r I Φ c for =, 3,.... Proof. The proof is given in section 0. Finally, we present two results related to stability of the coarse-grid correction. It is well nown that for the canonical restriction operator the inequality r c r holds with a constant c r independent of. The second stability result is the following. Theorem 6.5. There exists a constant c 3 independent of and such that 6.3 A p A c 3 for =, 3,.... Proof. The proof is given in section 9. We now obtain a two-grid convergence result. Theorem 6.6. For the two-grid method we then have T AT A c c ++c r c 3 d µ ν 4. Proof. The proof is based on results from 6.9, 6., 6., and 6.3. We use the splitting in 6.8. From the Assumptions 6. and 6. and Lemma 6. it follows that S,pr and S,po. From Assumption 6., Theorem 6.3, and Φ we obtain Φ Sµ,pr Φ S,pr Φ µ Φ d µ 4 Combine these bounds with the results in Corollary 6. and Theorems 6.4 and 6.5. Using the two-grid result of Theorem 6.6 we derive a multigrid W-cycle convergence result based on standard arguments. Theorem 6.7. In addition to the assumptions of Theorem 6.6 we assume that the number of smoothing steps on every grid level is sufficiently large: ν c po, µ c pr 4

18 78 MAXIM A. OLSHANSKII AND ARNOLD REUSKEN with suitable constants c po, c pr. Then for the contraction number of the multigrid W-cycle the inequality 6.4 M mgm A T A ξ holds, with a constant ξ < independent of and. Proof. Define ξ := M mgm mgm A T A. Using the recursion relation 6.6 for M it follows that ξ T A T A + S,po ν A p A ξ r S,pr µ T A T A + c 3c r ξ. Now use the two-grid bound given in Theorem 6.6 and a fixed point argument. Remar 6.4. We briefly discuss the arithmetic wor needed in one W-cycle iteration. The arithmetic wor for a matrix vector multiplication on level is of order ON =On. The wor needed in one smoothing iteration is of order ON. The number of smoothings behaves lie ν + µ 4. Using a standard recursive argument it follows that for a multigrid W-cycle iteration the arithmetic complexity is of the order N ln N 4. Hence this multigrid method has suboptimal complexity. 7. Proof of Theorem 6.. We recall the representation of the stiffness matrix in 3.4 A = + δ h A x + A y BD x. We will need the following lemma: Lemma 7.. The inequality BD x 0 holds. Proof. The matrix 6 BD x 3 h A x is the stiffness matrix corresponding to the bilinear form u, v Ω u xv dxdy. For any x X we get 6 BD xx, x 3 h A x x, x = x x P x dxdy = ΩP P x dxdy 0. Γ E Since the matrix A x is symmetric positive definite the result now follows. We now consider the preconditioner W = 4 I + D h x, as in 6.4. Theorem 7. =Theorem 6.. The inequality W A 8 I holds. Proof. First note that h ˆDx ˆDT x = ˆD x + ˆD T x h, 0,...,0 T, 0,...,0 ˆD x + ˆD T x and thus h ˆDT x ˆDx ˆDT x ˆDx ˆD T x ˆD x + ˆD T x ˆD x holds. Using Âx = ˆD T x ˆD x this results in h  x ˆD T x Âx and thus 7. h A x Dx T A x. Note that the following inequality holds for any a, b, c R and σ,σ,σ 3 > 0: a + b + c + σ + σ 3 a ++σ 3 + σ b ++σ + σ c.

19 MULTIGRID FOR A CONVECTION-DOMINATED PROBLEM 79 We apply this inequality with σ =,σ = σ 3 =. Also using A y 4h B 6 we get for any x X 7. A x 4 A y x +3 A x x BD xx 6 A y x, x +3 h A x x + 5 D xx. and We recall that = δh 7 6 h. Now apply the result 7. and the estimates in Lemmas 3. and Lemma 7. to obtain 4 W x, A x = h x + D x x, A y x + A x x + 6 BD xx 4 A y x, x + D x x, A x x + D x x, 6 BD xx h 4 = 8 8 h 3 6 A y x, x A x x + 3 D xx A y x, x + 4 h 7 A x x D xx A y x, x +3 A x x + 5 D xx. h Combination of this with the inequality in 7. proves the theorem. 8. Proof of Theorem 6.3. We start with an elementary nown result on the convergence of basic iterative methods. Lemma 8.. Assume C, A, W R n n with C symmetric positive definite. If there are constants c 0 > 0, c such that 8. c 0 Ay, Ay C Wy,Wy C c Wy,Ay C for all y R n then for arbitrary d [0, ] we have I α c 0 AW C d c 0 c c if d α + d. Proof. Let D := AW. From 8. we get Note that Dy, y C c y, y C, Dy, Dy C c 0 y, y C for all y. I α c 0 c AW y C and α α d if d α + d. = y, y C α c 0 Dy, y C + α c 0 c c Dy, Dy C α c 0 c + α c 0 c y C = α α c 0 c y C

20 80 MAXIM A. OLSHANSKII AND ARNOLD REUSKEN Below we use the scalar product, Φ := Φ, with Φ defined in 6.7. We recall the result proved in Lemma 5.3, 8. A x, D x x Φ c D x x Φ for all x X with c>0 independent of and of. We introduce the diagonal projection matrix J := I n Ĵ with Ĵ the n n diagonal matrix with Ĵ i,i = if ˆΦ i,i = and Ĵ i,i = 0 otherwise. Lemma 8.. There exists a constant c>0 independent of and such that W x Φ c h 3 I J x Φ + D x x Φ for all x X. Proof. Note that J x Φ = J D x J D x x Φ J Dx J Φ D x x Φ = J D x J D x x Φ 3 +h D x x Φ. And thus, using h we get W x Φ = 4 h x + D x x 4 Φ h I J x Φ + 4 h J x Φ + D x x Φ 4 4 h I J x Φ + c D x x Φ c h I J x Φ + D x x Φ. Squaring this result and using h completes the proof. h 3 We define ˆΦ x := h diagφ i φ i+ i n with φ i = φ ξ i as in 6.7. Consider the diagonal matrix Φ x := I n ˆΦ x. Note that Φ x 0. Lemma 8.3. The following estimate holds: A x, x Φ 30 Φ x x for all x X. Proof. Recall 8.3 A = A x + A y + 6 BD x. Note that 8.4 Φ A y =I n ˆΦ Ây Ĵ =Ây ˆΦ Ĵ 0. We consider the term Φ A x = I n ˆΦ Â x. Note that ˆΦ Â x = ˆΦ ˆDT x ˆDx.A simple computation yields ˆΦ ˆDT x ˆD x T ˆΦ = ˆΦ x ˆT, with ˆT := tridiag0, 0,, and thus 8.5 ˆΦ Â x = ˆDT x ˆΦ ˆDx ˆΦx ˆT ˆDx. From the Cauchy Schwarz inequality it follows that 8.6 ˆΦ x ˆT ˆDx y, y 9 4 ˆΦ x ˆT ˆD x y + 9 ˆΦ x y for all y R n.

21 MULTIGRID FOR A CONVECTION-DOMINATED PROBLEM 8 Using the property 5.5 we get ˆT T ˆΦx ˆT 8.7 c ˆΦ 0. Combination of the results in 8.5, 8.6, 8.7 and using c yields ˆΦ Â x y, y ˆD 9 x y ˆΦ 4 c 0 ˆD x y ˆΦ 9 ˆΦ x y And thus 9 ˆΦ x y for all y R n. 8.8 Φ A x 9 Φ x holds. Finally we consider the term 6 BD xx, x Φ. First we note BD x = bloctridiag ˆD x, 4 ˆD x, Ŝx, Ŝ x := h Rn n 0 and thus K := 6 Φ BD x = 6 bloctridiagˆφ ˆDx, 4ˆΦ ˆDx, ˆΦ Ŝ x. Hence K + KT = bloctridiagˆφ ˆDx + ŜT ˆΦ x, 4ˆΦ ˆDx + ˆD x T ˆΦ, ˆΦ Ŝ x + ˆD x T ˆΦ. A simple computation yields 8.9 ˆΦ ˆDx + ˆD x T ˆΦ = ˆΦ x + tridiag φ i,φ i + φ i+, φ i+ i n =: h ˆΦ x + R and ˆΦ Ŝ x + ˆD x T ˆΦ = ˆΦ x ˆT + h φ n e n e T n, with n := n and e n the nth basis vector in R n. Thus we obtain K + KT = bloctridiag T ˆT ˆΦx, 4ˆΦ x, ˆΦ x ˆT By ˆΦ x Note that + bloctridiag h φ n e n e T n, 4R, bloctridiag ˆT T ˆΦx, 4ˆΦ x, ˆΦ x ˆT. φ n e n e T n h Φ x we denote the pseudoinverse of ˆΦ x Φ x. We then have Φ x K + K T Φ x bloctridiagˆφ T x ˆT ˆΦ x, 4I, ˆΦ x ˆT ˆΦ x ˆΦ T x ˆT ˆΦ x = ˆΦ x ˆT ˆΦ x = max i 3+ φi φ i φ i φ i+ And thus we get Φ x K + K T Φ x 4 e 8 I. Hence Φ BD x = K 6 e 8 Φx. = e 8..

22 8 MAXIM A. OLSHANSKII AND ARNOLD REUSKEN Combination of the results in 8.3, 8.4, 8.8, and 8.0 yields Φ A e 8 Φ x > 30 Φ x. Using the previous two lemmas we can show a result as in the second inequality in 8.. Theorem 8.4. There exists a constant c independent of and such that W x, W x Φ c W x, A x Φ for all x X. Proof. From Lemma 8.3 and 8. we get 8. W x, A x Φ = 4 h x, A x Φ + D x x, A x Φ c h Φ x x, x + D x x Φ with c>0 independent of and. Using φ i φ i+ = e 4 φ i 5 φ i for i 3 + we get 8. Φ x x, x 5 h I J Φ x, x = 5 h I J x Φ. From 8. and 8. we obtain W x, A x Φ c h 3 I J x Φ + D x x Φ Now combine this with the result in Lemma 8.. We now consider the first inequality in 8.. Theorem 8.5. There exists a constant c 0 > 0 independent of and such that c 0 A x, A x Φ W x, W x Φ for all x X. Proof. The constants c that appear in the proof are all strictly positive and independent of and. First note that A x Φ A x x Φ + A y x Φ + 6 BD xx Φ. We have A y Φ = I n ˆΦ Ây ĴI n ˆΦ = Ây Ĵ 4 h. Note that φ i φ i+ e 4 and thus ˆΦ ˆDT x ˆΦ ch holds. From this it follows that Dx T Φ ch holds. With a similar argument we get B Φ c. Thus we obtain, using 3 h, 8.3 A x Φ Dx T Φ D x x Φ + 4 h x Φ + c D x x Φ c h x Φ + D x x Φ.

23 MULTIGRID FOR A CONVECTION-DOMINATED PROBLEM 83 From 8.9 it follows that D x x, x Φ 0 holds. Using this we get 8.4 W x Φ = 6 h 4 x Φ + 6 h D x x, x Φ + D x x Φ c h 4 x Φ + D x x Φ. Now combine 8.3 with 8.4. Combination of the results of Theorems 8.4 and 8.5 with the second result in Lemma 8. shows that Theorem 6.3 holds. 9. Proof of Theorem 6.5. Let g X be given and define g := P g V. Let u V be such that a u,v =g,v for all v V. Then A g = P u holds. The corresponding continuous solution u V satisfies a u, v =g,v for all v V. Now note that A p A g A p P = max u,y y X y 9. c max v V a u,v v a u,v a u,v a u,v c max + c max. v V v v V v Define e := u u. For the first term in 9. we get, using the results of Lemma 4.3, a u,v a e,v + a u, v ch e x v x + e y v y + e x v + g,v c e x + e y v + g v h 9. c g v c g v. For the second term in 9. we have, using Lemma 4., 9.3 a u,v a u,v = δh u x, v x c u x v c g v c g v. Combination of the results in 9., 9., and 9.3 yields A p A g c g and thus the result in Theorem 6.5 holds. 0. Proof of Theorem 6.4. We briefly comment on the idea of the proof. As usual to prove an estimate for the error in the L -norm we use a duality argument. However, the formal dual problem has poor regularity properties, since in this dual problem Γ E is the inflow boundary and Γ W is the outflow boundary. Thus Dirichlet outflow boundary conditions would appear and we obtain poor estimates due to the poor regularity. To avoid this, we consider a dual problem with Neumann outflow and Dirichlet inflow conditions. To be able to deal with the inconsistency caused by these wrong boundary conditions we assume the right-hand side is zero

24 84 MAXIM A. OLSHANSKII AND ARNOLD REUSKEN near the boundary Γ W. In order to satisfy this assumption we use the cut-off operator with matrix Φ. A further problem we have to deal with is the fact that due to the level dependent stabilization term we have to treat -dependent bilinear forms. We introduce the space V 0 := {v V v x = 0 for all x Ω in }. Let ˆb X be given. In view of Theorem 6.4 we must prove an estimate W A p A r I Φ ˆb c ˆb with a constant c that is independent of,, and ˆb. Note that P I Φ ˆb =: f V 0 holds. For this f V 0 we define corresponding discrete solutions and continuous solutions as follows: 0. u V : a u,v =f,v for all v V, u V : a u, v =f,v for all v V, u V : a u,v =f,v for all v V, ũ V : a ũ, v =f,v for all v V. In the proof of Lemma 4. we showed that v x = D x P v holds for all v V. We use that W = 4 I + D h x and obtain W A p A r I Φ ˆb 4 A p A r I Φ ˆb 0. + D x A I Φ ˆb + D x p A r I Φ ˆb c h u u + u x + u x c u u + ũ u + u ũ + u x + u x. h From Lemma 4. we get h 0.3 u x + u x c f. From the result in Theorem 0. below it follows that u u + u ũ c h 0.4 f. Finally, from Theorem 0.4 we have 0.5 u ũ ch f. If we insert the results 0.3,0.4, and 0.5 in 0. we get W A p A r I Φ ˆb c f c P I Φ ˆb c ˆb and thus the result of Theorem 6.4 is proved. It remains to prove the results in Theorems 0. and 0.4. Theorem 0.. For f V 0 let u and u be as defined in 0.. Then 0.6 holds. u u c h f

25 MULTIGRID FOR A CONVECTION-DOMINATED PROBLEM 85 Proof. Define e := u u. Let w H Ω be such that 0.7 w yy w xx w x = e with 0.8 w x = 0 on Γ W, w = 0 on Γ \ Γ W. Note that for this problem Γ E is the inflow boundary and Γ W is the outflow boundary. We multiply 0.7 with e and integrate by parts to get e = e y,w y + e x,w x Γ E w x e dy +e x,w = a e,w Γ E w x e dy. We use 4.6 with w and e instead of u and f, respectively, and 4. to estimate w x e dy wx Γ E Γ dy e dy ch e h 0.9 f. E ΓE From this estimate and the Galerin orthogonality for the error it follows that for any v V 0.0 e ey, w vy+ ex, w vx +e x,w v +c e h 3 f. Let Ω h := Ω h be as defined in 5, i.e., Ω h is the set of triangles with at least one vertex on Γ W. In the remainder of the domain, ω =Ω\Ω h, we tae v as a nodal interpolant to w and we put v = 0 on Γ W to ensure v V. Note that v is a proper interpolant of w everywhere in Ω except in Ω h. Therefore we will estimate scalar products in 0.0 over ω and Ω h, separately. We continue 0.0 with e ch e y ω w H ω + c h e x ω w H ω +ch e x ω w H ω + c e h 3 f +I Ωh 0. ch f e +I Ωh. The term I Ωh collects integrals over Ω h : I Ωh = e y, w v y Ωh + e x, w v x Ωh +e x,w v Ωh. To estimate I Ωh we use Corollary 5.7 and the following auxiliary estimate for the interpolant v V of w, with ω h = {x, y Ω: x h, h }: v Ωh c v ωh c w ωh + v w ω h [ x = c w0,y+ w η η, y dη] dx dy 0 h c h w dy + h w x + h w H ω Γ W 0 + v w ω c h + h e.

26 86 MAXIM A. OLSHANSKII AND ARNOLD REUSKEN We proceed estimating terms from I Ωh, where we use the previous result: e y, w v y Ωh e y Ωh w y + v y Ωh c h f e + h v Ωh c h f + h + h e c h + h f e, e x, w v x Ωh e x Ωh w x + v x Ωh ch f e + h v Ωh c h + h 5 f e, e x,w v Ωh e x Ωh w Ωh + v Ωh ch f h w dy + h w x Ωh + v Ωh Γ W c h + h 5 f e. Inserting these estimates into 0. and using h we obtain e c h f e + c h + h + h 5 f e c h f e. and thus the theorem is proved. For the proof of Theorem 0.4 we first formulate two lemmas. Lemma 0.. Consider a function g H Ω. The solution of 0. u xx u yy + u x = g x with boundary conditions as in. satisfies 0.3 u dy c h g + g dy + h g x. Γ E Γ E 0.4 Proof. We multiply 0. with u and integrate by parts to get u x + u y + u dy = g, u x + g u dy. Γ E Γ E For the right-hand side in 0.4 we have g, u x g u x c g g x c h g + h g x and gudy g dy + u dy. Γ E Γ E 4 Γ E Combining these estimates and 0.4 the lemma is proved.

27 MULTIGRID FOR A CONVECTION-DOMINATED PROBLEM 87 Lemma 0.3. Assume g H and g ΓE =0, let u be the corresponding solution of 0.. Then the following holds: u c g + h g x + g dy + h u 0.5 x dy. Γ W Γ W Note that the standard a priori estimates would give only u c g x. Proof. Consider the auxiliary function vx, y := x uξ,y dξ. It satisfies v xx v yy + v x = g + u in + g in, with u in x, y =u x 0,y and g in = g0,y. The corresponding boundary conditions are 0.7 v x = u,yonγ E, v =0on Ω \ Γ E. Then the estimate 0.5 is equivalent to v x c g + h g x + g dy + h u 0.8 x dy. Γ W Γ W The estimate 0.8 is proved by the following arguments. We multiply 0.6 with v x and integrate by parts to obtain v x + v y dy + v x dy Γ E Γ W =g, v x + u in,v x +g in,v x v x dy. Γ E Since g ΓE = 0 the estimate 0.3 yields 0.0 v x dy = u dy c h g + h g x. Γ E Γ E Now 0.8 follows from 0.9 by applying the Cauchy inequality and estimate 0.0. Using these lemmas we can prove the final result we need. Theorem 0.4. For f V 0 let u and ũ be the continuous solutions defined in 0.. Then the following holds: 0. u ũ ch f. 0. Proof. The difference e := u ũ solves the equation e xx e yy + e x = g x, with g = δh ũ x and boundary conditions as in.. Now the result of Lemma 0.3 can be applied. We obtain e c g + h g x + g dy + h e x dy Γ W Γ W ch ũ x + h ũ xx + u x dy + ũ x dy. Γ W Γ W