c 2011 Society for Industrial and Applied Mathematics

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1 IAM J NUMER ANAL Vol 49, No 2, pp c 2011 ociety for Industrial and Applied Mathematics FIELD-OF-VALUE CONVERGENCE ANALYI OF AUGMENTED LAGRANGIAN PRECONDITIONER FOR THE LINEARIZED NAVIER TOKE PROBLEM MICHELE BENZI AND MAXIM A OLHANKII Abstract We study a block triangular preconditioner for finite element approximations of the linearized Navier tokes equations The preconditioner is based on the augmented Lagrangian formulation of the problem and was introduced by the authors in [IAM J ci Comput, ), pp ] In this paper we prove field-of-values type estimates for the preconditioned system which lead to optimal convergence bounds for the GMRE algorithm applied to solve the system Two variants of the preconditioner are considered: an ideal one based on exact solves for the velocity submatrix, and a more practical variant based on block triangular approximations of the velocity submatrix Key words Navier tokes equations, augmented Lagrangian, preconditioning, field of values, generalized minimal residual AM subject classifications 65F10, 65N22, 65F50 DOI / Introduction Let Ω R d be a bounded domain with sufficiently smooth boundary Consider the Oseen equations in Ω, αu νδu +a ) u + p = f, 11) div u =0, together with suitable boundary conditions Problem 11) arises from the linearization of the unsteady Navier tokes equations describing a viscous, Newtonian incompressible fluid with kinematic viscosity ν>0 The unknown quantities in 11) are the velocity field u and the pressure p; the forcing term f, which represents external forces such as gravity), is given The parameter α is positive and proportional to the reciprocal of the time step for unsteady problems; it is zero in the steady case We assume that the convection coefficient a is sufficiently smooth and divergence-free div a = 0) For the purpose of theoretical analysis we will also assume, without loss of generality, that a L =1 Considerable effort has been devoted to the development of efficient solvers for linear systems arising from discretizations of problem 11) uch linear systems are of saddle point type and require specialized techniques for their efficient and robust solution everal preconditioners for Krylov subspace methods have been proposed and analyzed; see [1] and [9] for broad overviews of this active field of research Recently, a promising approach based on the augmented Lagrangian AL) formulation [10] has been proposed in [2] and further studied in [3, 4, 16]; see also [18] The Received by the editors August 25, 2010; accepted for publication in revised form) January 4, 2011; published electronically April 14, Department of Mathematics and Computer cience, Emory University, Atlanta, GA benzi@mathcsemoryedu) The work of this author was ported in part by a grant of the University Research Committee of Emory University Department of Mechanics and Mathematics, Moscow tate M V Lomonosov University, Moscow , Russia MaximOlshanskii@mtu-netru) The work of this author was ported in part by the Russian Foundation for Basic Research grants , and by a Visiting Fellowship granted by the Emerson Center for cientific Computation at Emory University 770 Copyright by IAM Unauthorized reproduction of this article is prohibited

2 FOV ANALYI OF AUGMENTED LAGRANGIAN METHOD 771 preconditioners studied in these papers are of block triangular type The extensive numerical experiments reported in these papers indicate that such preconditioners result in h-independent convergence where h denotes the discretization parameter) and rather good robustness even for very small viscosity ome theoretical results have been given in [2] and [3] in the form of eigenvalue bounds for the preconditioned matrices While these bounds are useful, for instance, in the choice of the augmentation parameter see [4]), they do not say much about the convergence behavior of preconditioned Krylov subspace methods such as GMRE [17] Indeed, as is well known, knowledge of the eigenvalues alone is generally not sufficient to characterize the convergence of Krylov iterations applied to nonnormal matrices, and it may even be misleading in some cases In this paper we make use of field-of-values FOV) estimates [14, 15] to prove the observed h-independent performance of block triangular preconditioners for the AL formulation of the discrete Oseen problem In the ideal case where linear systems associated to the velocity submatrix in the preconditioner are solved exactly, we prove that preconditioned GMRE converges at a rate independent of all problem parameters h, ν, andα For the modified inexact) variant of the preconditioner introduced in [3] we establish h-independent convergence of GMRE The remainder of the paper is organized as follows In section 2 we make some assumptions and preliminary observations on the finite element FE) approximation of 11) The algebraic problem, its AL formulation, and the block triangular preconditioner are described in section 3 ome useful bounds for the chur complement of the original saddle point system are derived in section 4 The main results are given in section 5 for the ideal version of the preconditioner and in section 6 for the modified one ection 7 contains some closing remarks 2 The finite element approximation Let {T h } h>0 be a family of triangulations of Ω in the sense of [6] For the sake of analysis we assume the standard regularity condition to be satisfied by {T h }: diamk) max C<, h>0 K T h ρk) where ρk) is the radius of the ball inscribed in K Note that we allow locally refined meshes We assume conforming FE spaces V h H 1 0 Ω) and Q h L 2 0 Ω), satisfying the LBB stability condition div v h,q h ) 21) inf 1 q h Q h v h V h v h q h In order to avoid the repeated use of generic but unspecified constants, here and in the remainder of the paper the binary relation x y means that there is a constant c such that x cy,andcdoes not depend on the parameters which x and y may depend on, eg, ν, α, andh Obviously, x y is defined as y x, andx y when both x y and y x For α, 0 consider the following bilinear form on V h Q h ) V h Q h ): au h,p h ; v h,q h ):=αu h, v h )+ν u h, v h )+P h div u h, div v h ) +a ) u h, v h ) p h, div v h )+q h, div u h ), where P h is the L 2 orthogonal projector from L 2 into Q h The FE method for the Oseen problem is based on the weak formulation: Find {u h,p h } V h Q h solving 22) au h,p h ; v h,q h )=f, v h ) {v h,q h } V h Q h Copyright by IAM Unauthorized reproduction of this article is prohibited

3 772 MICHELE BENZI AND MAXIM A OLHANKII Due to the obvious equalities 23) P h div u h, div v h )=P h div u h,p h div v h ), p h, div v h )=p h,p h div v h ), q h, div u h )=q h,p h div u h ), the solution of 22) does not depend on, although the coefficient matrix of the system does For further study of the algebraic properties of the system we need some results from FE analysis To this end, we introduce the following scalar products and norms based on the symmetric part of au h,p h ; v h,q h )): u h, v h ) a := αu h, v h )+ν u h, v h )+P h div u h, div v h ) on V h, {u h,p h }; {v h,q h }) a := u h, v h ) a +ν + )p h,q h ) on V h Q h Lemma 21 The bilinear form au h,p h ; v h,q h ) satisfies the following stability and continuity estimates: 24) 25) au h,p h ; v h,q h ) inf C 1, {u h,p h } V h Q h {v h,q h } V h Q h {u h,p h } a {v h,q h } a au h,p h ; v h,q h ) C 2, {u h,p h } V h Q h {v h,q h } V h Q h {u h,p h } a {v h,q h } a with some positive mesh-independent constants C 1 1+ν + ) ν + + α + 1 )) 1, C 2 ν + α ν 1 2 ν α 1 2 ) Proof Consider the auxiliary norm with a parameter σ>0 such that {u h,p h } σ := u h 2 a + σ p h 2 ) 1 2 on V h Q h, 26) σ ν + + α +ν + α) ) Due to Lemma 31 in [11] there exists a constant σ satisfying 26) such that 27) inf {u h,p h } V h Q h {v h,q h } V h Q h au h,p h ; v h,q h ) {u h,p h } σ {v h,q h } σ 1 Thus the lower bound in 24) follows from 27) and {v h,q h } a max{1, ν + ) 1 2 σ 1 2 } {vh,q h } σ v h,q h V h Q h The estimate 25) follows from the definition of a, ;, ) and 23) through the application of the Cauchy chwarz and Friedrichs inequalities in a straightforward Copyright by IAM Unauthorized reproduction of this article is prohibited

4 FOV ANALYI OF AUGMENTED LAGRANGIAN METHOD 773 way: au h,p h ; v h,q h ) α u h v h + ν u h v h + P h div u h P h div v h + a L u h v h + p h P h div v h + q h P h div u h α u h v h + ν u h v h + P h div u h P h div v h 1 + ν 1 2 ν α 1 2 ) ν 1 2 uh α 1 2 vh + C F ν 1 2 vh ) +ν + ) 1 2 ph 1 2 Ph div v h + ν 1 2 vh ) +ν + ) 1 2 qh 1 2 Ph div u h + ν 1 2 uh ) 1 ν 1 2 ν α 1 2 ) {u h,p h } a {v h,q h } a, where C F denotes the constant in the Friedrichs inequality 3 Algebraic problem and preconditioning Let {φ i } 1 i n and {ψ j } 1 j m be bases of V h and Q h, respectively Define mass matrices for velocity and pressure elements, and the matrices M u ) i,j =φ j,φ i ), M p ) i,j =ψ j,ψ i ), D i,j = φ j, φ i ), N i,j =a ) φ j,φ i ), B i,j = div φ j,ψ i ) We shall use the following notation: A := αm u + νd + N, A := A + B T W B, with an auxiliary m m matrix W, assumed to be symmetric and positive definite With this notation and the natural ordering of unknowns, the linear algebraic system corresponding to the FE problem 22) takes the following form for =0: ) ) ) A B T u f 31) =, or A x = b B 0 p 0 The augmented Lagrangian AL) approach from [2] consists first of replacing the original system 31) with the equivalent one ) ) ) A + B 32) T W B B T u f =, or A B 0 p 0 x = b, followed by preconditioning 32) with a block triangular preconditioner of the form )  B 33) P = T 0 1 ν+ W Hereandinwhatfollows denotes a preconditioner for the velocity block A A typical choice for W is a diagonal approximation of M p ; see [2] Using the main diagonal of M p works well in practice [19] Remark 31 For W = M p the matrix A is the coefficient matrix of the FE problem 22) for any Copyright by IAM Unauthorized reproduction of this article is prohibited

5 774 MICHELE BENZI AND MAXIM A OLHANKII As was mentioned in the introduction, eigenvalue bounds for P A have been established in [2] and [3] The present paper is concerned with FOV-type bounds for A which lead to rigorous convergence estimates for GMRE For this purpose we need a few more technicalities introduced below For a symmetric positive definite matrix H we define the H-inner product of two vectors u and v by u, v H := Hu,v, with the corresponding vector and induced matrix norm Given two symmetric positive definite matrices H 1 R n1 n1 and P H 2 R n2 n2 and a matrix C R n2 n1, we will use the following notation for the norm of C as an operator from R n1 to R n2 : 34) C H1 H 2 := x R n 1 Cx H2 x H 1 Let A := 1 2 A + A T ) Note that N = N T,soA = αm u + νd + B T W B is a symmetric positive definite matrix and therefore invertible We need a few specific scalar products and corresponding norms: p, q M := M p p, q on R m, {u, p}, {v, q} a := A u, v +ν + ) M p p, q on R n+m, {u, p}, {v, q} a := A u, v +ν + ) Mp p, q on R n+m The corresponding norms on R n+m are denoted by z a = {u, p} a and z a = {u, p} a for z = {u, p} We note that {u, p}, {v, q} a ={u h,p h }; {v h,q h }) a, where u, v, p, q are vectors of coefficients for FE functions u h,p h, v h,q h and a is the dual norm of a with respect to the l 2 -duality Due to Remark 31, the result of Lemma 21 is equivalent to the following conditions for the augmented matrix A : 35) 36) inf z 1 R n+m z 1 R n+m A z 1, z 2 C 1, z 2 R z n+m 1 a z 2 a A z 1, z 2 C 2, z 2 R z n+m 1 a z 2 a with the same constants C 1 and C 2 as in 24) and 25) For B R n+m) n+m) denote μb) := Bz, z inf a z R n+m z, z a The convergence of the preconditioned GMRE method in the product norm can be estimated as [7] 37) r k a 1 μa P ) μpa )) k/2 r 0 a, where r k is the kth residual vector from R n+m 4 ome bounds for 0 In this section we prove and discuss some bounds for the chur complement of the original nonaugmented) saddle point problem, 0 := BA B T A = A for =0), and its inverse These results are important for our further analysis We note that in some situations eg, in the case of enclosed flow) the discrete divergence operator B Copyright by IAM Unauthorized reproduction of this article is prohibited

6 FOV ANALYI OF AUGMENTED LAGRANGIAN METHOD 775 does not have full row rank, making 0 singular In this case, 0 is to be understood as the Moore Penrose generalized inverse 0 imilarly, when taking the infimum of the generalized Rayleigh quotient 0 q, q / q 2 M, we consider only vectors q in the orthogonal complement of the null space of 0 In order to keep the notation simple, these conventions will be tacitly assumed in what follows We begin with the following lemma Lemma 41 The following bounds hold: 0 inf M pq, q M 41) q R m q 2 ν, M 42) 0 q, q 1 inf q R m q 2 M α + ν + ν Proof First we show 41) For arbitrary q R m,considerp = 0 M pq This can be rewritten as BA B T p = M p q Bu = M p q, Au = B T p Therefore the FE counterparts q h Q h of q and p h Q h of p satisfy the following relations for all r h Q h, v h V h, and an auxiliary u h V h : q h,r h ):= div u h,r h ), 43) αu h, v h )+ν u h, v h )+a )u h, v h )= p h, div v h ) We set in 43) v h = u h and use a )u h, u h )=0toget 0 M pq, q M = M pp, q =p h,q h )= div u h,p)=ν u h 2 + α u h 2 ν div u h 2 = ν q h 2 = ν q 2 M, which implies 41) The estimate in 42) is proved below with the help of the following identity [8]: 1 44) A + A T ) = A I 1 1 A 2 NA 2 )2 ) A 1 2 Recall that A = A + N, A = αm u + νd is the symmetric part of A discretization of νδ+αi), and N is the skew-symmetric part of A discretization of a )) To show 42) we first note the estimate 45) BA BT q, q 2 Bv,q = v R A n+m v, v = div v h,q h ) 2 v h V h ν v h 2 + α v h 2 1 A 2 BT q, A 1 2 BT q 1 + A NA 2 )2 ) 1 ν + α)1 + A 1 2 div v h,q h ) 2 v h V h ν + α) v h 2 1 ν + α q h 2 = 1 ν + α q 2 M With the help of 44) and 45) we obtain 0 q, q = AB T q, B T q = I A NA 2 )2 ) A 1 2 BT q, A 1 2 BA = BT q, q 46) 1 + A NA 2 )2 ) 1 NA 2 q 2 M )2 ) BT q Copyright by IAM Unauthorized reproduction of this article is prohibited

7 776 MICHELE BENZI AND MAXIM A OLHANKII Further, we estimate 47) A NA 2 )2 A NA 2 2 =max{ λ 2 : λ spa NA 2 )} =max{ λ 2 : λ spa N)} A N 2 M u, wherewehaveusedsp ) to denote the spectrum For a given w R n and u = A Nw consider their FE counterparts w h, u h V h, respectively Then it holds that ν u h, v h )+αu h, v h )=a )w h, v h ) v h V h etting v h = u h and applying the Cauchy chwarz and Young inequalities, we get ν u h 2 + α u h 2 =a )w h, u h )= a )u h, w h ) ν 2 u h 2 + ν a 2 L w h 2 Hence, by the Friedrichs inequality and a L =1,weobtain ν + α) u h 2 ν u h 2 + α u h 2 ν w h 2 Therefore u h 2 1 αν + ν 2 w h 2, and we have proved 48) A 1 N 2 M u αν + ν 2 Estimates 46), 47), and 48) yield the estimate in 42) The best constant C from the estimate 49) 0 M p M C plays an important role in our convergence analysis Below we prove some theoretical bounds for this constant and perform a few numerical experiments to assess the actual value of C and check its dependence on problem parameters The lemma below gives the most general bound, which is valid for general convection fields, boundary conditions, and FEs; however, it can be nonoptimal in terms of ν and α Lemma 42 The constant C satisfies 410) C ν + α + 1 ν + α Proof For arbitrary q R m and p = 0 M pq it holds that p M q M = p h q h, where q h Q h is the FE counterpart of q and p h Q h together with some auxiliary u h V h satisfies 43) for all r h Q h, v h V h With the help of the LBB condition Copyright by IAM Unauthorized reproduction of this article is prohibited

8 FOV ANALYI OF AUGMENTED LAGRANGIAN METHOD ) and the Cauchy chwarz inequality and using a )u h, v h )=a )v h, u h ) and a L = 1, we get from the second relation in 43) p h, div v h ) 411) p h α +1) u h + ν u h v h V h v h The LBB condition can be reformulated as follows: For a given q h there exists w h V h such that 412) w h q h and div w h,r h )=q h,r h ) r h Q h Further decompose u h = w h + u 0 h Relations 43) yield for all r h Q h, v h V h 413) div u 0 h,r h)=0, αu 0 h, v h)+ν u 0 h, v h)+a )u 0 h, v h)+p h, div v h ) = αw h, v h )+ν w h, v h )+a )w h, v h ) etting v h = u 0 h in the second relation of 413) and using the first one, we obtain α u 0 h 2 + ν u 0 h 2 = αw h, u 0 h)+ν w h, u 0 h)+a )w h, u 0 h) With the same arguments as in 411) and applying 412) and the Friedrichs inequality, we estimate α u 0 h 2 + ν u 0 h 2 α w h 2 + ν w h 2 + a L ν + α w h 2 414) ν + α + 1 ) q h 2 ν + α Combining 412) and 414), we get through the triangle inequality 415) α u h 2 + ν u h 2 ν + α + 1 ) q h 2 ν + α Now 411) and 415) give p h ν + α + 1 ) q h, ν + α which completes the proof Next, we argue that in general the bound in 410) can be pessimistic for the case of α 0 Indeed, assume that the problem admits Fourier analysis, ie, a is a constant vector and periodic boundary conditions are imposed; then the action of 0 on a given harmonic ψ k =expık x) is 0 ψ k =ν + α k 2 + ı a k) k 2 )ψ k Thus, in such a setting it holds that 0 2 = λ max 0 T 0 )=max {ν + k >0 α k 2 ) 2 +a k) 2 k 4 } Therefore, we get 416) C ν + α +1, which does not blow up for ν 0andα 0 in contrast to 410) Copyright by IAM Unauthorized reproduction of this article is prohibited

9 778 MICHELE BENZI AND MAXIM A OLHANKII Fig 1 The driven cavity vortex given by 418) It is interesting to point out that the ν, α-dependence scenario from 416) can be shown to hold in somewhat more general cases than the periodic one For example, 416) can be proved to hold on the continuous level in the case of mixed boundary conditions in a bounded domain: u n = 0, u) n = 0on Ω However, a is still assumed to be a constant vector Below we show results of experiments where we have computed 0 M p M for a set of different parameters and convection fields a One example of a is the Poiseuille flow in a channel given by { 4y1 y), 417) ax, y) = 0 The second example is the driven cavity -type vortex [5]: ) r 2 expr 2y) 2π expr sin 2πexpr2y)) 2)) expr 2) 1 cos 418) ax, y) = ) r1 expr 1x) 2π expr sin 2πexpr1x)) 1)) expr 1) 1 cos The position of the center of the vortex, 2πexpr1x)) expr 1) 2πexpr2y)) expr 2) x 0,y 0 )=r 1 logexpr 1 )+1)/2), r 2 logexpr 2 )+1)/2)), )), )) is governed by the two parameters r 1 and r 2 Unless otherwise stated we set r 1 =4 and r 2 =01; the generated velocity field is shown in Figure 1 To produce numerical results we use two FE pairs, one with continuous pressure approximation and the other with discontinuous pressure approximation These are, respectively, the isop2-p1 and isop2-p0 FE pairs Both are LBB stable in the sense of 21); see [13] We use the uniform west-north triangulation of Ω = 0, 1) 2 Values of h in the tables below are 2 times the radius of the pressure element, while the velocity is piecewise linear with respect to a two times finer grid Note that A is a dense matrix, and so is 0 R m m Hence, the computational cost for finding 0 and 0 M p M quickly becomes prohibitive with increasing number of unknowns, restricting our calculations to rather moderate values of h etting h = 1/32 results in m = 1089, n = 7938 isop2-p1) and m = 2048, n = 7938 isop2-p0) The values of 0 M p M for both choices of a and FEs are given in Tables 1 and 2 We put α = 0 and vary ν from 1 down to 1e-7 For these test problems the Copyright by IAM Unauthorized reproduction of this article is prohibited

10 FOV ANALYI OF AUGMENTED LAGRANGIAN METHOD 779 Table 1 Values of 0 Mp M for isop2-p1 FEs Mesh size h Viscosity ν a is the Poiseuille flow 417) 1/ / a is the rotating vortex 418) 1/ / Table 2 Values of 0 Mp M for isop2-p0 FEs Mesh size h Viscosity ν a is the Poiseuille flow 417) 1/ / a is the rotating vortex 418) 1/ / results suggest that 0 M p M and thus C are of order 1 except for some growth for the case of the Poiseuille flow and extremely small viscosities It is important to note that the results for such small values of ν are purely of academic interest, since in practice such flows no longer have an equilibrium state and time-stepping schemes have to be adopted for the computations, leading to α>0 Also, for large mesh Reynolds numbers a stabilization is often added in practice We do not study how these additional mesh-dependent stabilizing terms affect the results In general, such terms add some extra numerical diffusion; therefore we do not expect any substantial growth of 0 M p M in this case An interesting question is whether high gradients of a affect the values of C To experiment with this we perform a series of experiments with a given in 418), for increasing values of the parameter r 1 Increasing r 1 has the effect of driving the vortex closer to the right boundary of Ω, forming a boundary layer ince the characteristic boundary layer is known to be of Oν 1 2 ) thickness, we set the corresponding value of ν equal to a L 1 r 1 logexpr 1 )+1)/2)) 2 Thefactor a L appears because we rescale the equations in such a way that the L norm of the convection field is 1 The results the computed 0 M p M and reference values of ν) are given in Table 3 We note that increasing r 1 further would not make sense, since a computationally affordable mesh size h would not resolve the layer 5 Analysis for exact AL approach In this section we analyze the ideal case, in which  = A Although not very practical, analyzing this case is useful as it provides a best case scenario in terms of convergence behavior of AL-based preconditioners Concerning the exact AL approach we prove the following main result Copyright by IAM Unauthorized reproduction of this article is prohibited

11 780 MICHELE BENZI AND MAXIM A OLHANKII Table 3 The values of 0 Mp M for the case of boundary layer in a, h =1/32 r ν 137e-1 285e-2 384e-3 481e-4 isop2-p0 FE 0 Mp M isop2-p1 FE 0 Mp M Theorem 51 Assume α + ν + ν For the FOV quantities in the GMRE convergence estimate 37) the following bounds hold: 51) + C 2 min{α,ν,ν }) + ν) 2 + C 2 1 μpa ), μa P ) Proof By the definitions of μ ) and matrices A and P, 51) is equivalent to 52) [ ][ ] [ ] I 0 u u ν + ) W BA ν + ) W, p p inf [u,p] R n+m and 53) inf [u,p] R n+m [ ][ I 0 u ν + ) W p BA a [u, p] 2 1 [u, p] 2 ] [ u, p + C2 min{α,ν,ν}) + ν) 2 + C 2 The key is to show the following estimates for the chur complement matrix of the augmented system: ] a 54) 55) ν + ) q 2 M M pq, q q M Rm, C 2 min{α,ν,ν} + ) C 2 + q 2 2 M q, q q R m Here and below we make use of the identity cf [12, Proposition 21]) 56) q, q 57) inf q R m q 2 = inf M q R m = 0 + W Using the spectral equivalence of W and M p, we obtain 54) from 56) and 41) Next, we show 55) Using 56), we find M pq, q M 0 M pq, q M q 2 M + q R m inf M p q 2 M M p q 2 M By the triangle inequality and the spectral equivalence of W and M p,weget 58) M pq 2 M = 0 M pq + W M p q 2 M 0 M pq 2 M + 2 q 2 M Copyright by IAM Unauthorized reproduction of this article is prohibited

12 FOV ANALYI OF AUGMENTED LAGRANGIAN METHOD 781 Estimates 57) and 58) imply q, q q 2 M inf q R m q 2 inf + 0 M pq, q M M q R m 2 q 2 M + 0 M pq 2 M 59) = inf q R m [ q M 0 Mpq,q M ] Mpq M 0 Mpq 2 M [ ] 2 2 q M +1 0 Mpq M The second term in the numerator on the right-hand side of 59) can be handled due to Lemma 41) as follows: 0 510) inf M pq, q M 0 q, q 1 q R m 0 M = inf pq 2 q R M m q 2 M α + ν + ν Now from 59) and 510) we get q, q 511) inf q R m q 2 inf M q R m [ ] 2 q M Mpq M α+ν+ν [ ] 2 2 q M +1 0 Mpq M Note that for α + ν + ν the function ft) = t+α+ν+ν ) 2 t+1 is monotonically increasing Therefore, based on 49) we finally get q, q 512) inf q R m q 2 C 2 +α + ν + ν ) M 2 C 2 +1 This proves 55) Owing to the definition of the product, a, together with the spectral equivalence of W and M p and the estimates 54) and 55) for the chur complement, the bounds in 52) and 53), and hence the theorem, will follow from the lower bounds [ ][ ] [ ] A 0 u u BA, p p 513) inf [ ][ ] [ ] [u,p] R n+m A 1 0 u u 0, p p and 514) inf [u,p] R n+m [ A ν + )W BA 0 ν + ) 2 W W ][ u p [ A 0 0 ν + ) 2 W W ][ ] [ ] u u, p p ] [ ] 1 u, p By simple eigenvalue estimates the quantities on the left-hand sides of 513) and 514) are easily shown to be greater than or equal to 1 2 ; see [14, page 586] Corollary 52 For α + ν + ν the residual norms in the GMRE method with ideal AL preconditioner satisfy r k a 1 c + C2 min{ν, ν,α ) k/2 }) + ν) 2 + C 2 r 0 a Copyright by IAM Unauthorized reproduction of this article is prohibited

13 782 MICHELE BENZI AND MAXIM A OLHANKII for some positive constant c independent of h and other parameters In particular, for the choice C this choice satisfies the restriction α + ν + ν due to Lemma 42), we get r k a q k r 0 a, where q < 1 is independent of h and other problem parameters The foregoing convergence result sheds some light on the optimal choice of Indeed, the eigenvalue analysis in [2] shows that 515) sp PA )={1} { + ν + μ i where μ i are the generalized eigenvalues of 516) BA B T q = μm p q } :1 i m, Then, applying estimates for μ i from [8] for the case α = 0), one concludes that it is sufficient to set = Oν ) to ensure that all nonunit eigenvalues of P  are contained in a box [a, A] [b, B], a > 0, in the complex plane, with a, A, b, B independent on ν and h However, all numerical results we are aware of suggest that = O1) is already sufficient for parameter-independent convergence The present analysis shows that the overestimation comes from applying the eigenvalue bounds of [8], which are likely nonoptimal Within our analysis the same choice of = Oν ) appears only if one assumes the bound of C from Lemma 42 to be optimal, which may not be true for many flows of interest, as suggested by the Fourier analysis and numerical experiments of section 4 6 Analysis for modified AL preconditioner We need some further notation and definitions Consider the natural block structure of A for simplicity we consider the two-dimensional case, although the results below hold for the threedimensional case as well): ) A11 A A = T 12 A 12 A 22 We define the preconditioner  B P = T 0 ρ ν+ W ), where ) A11 A  = T 12 0 A 22 and ρ is an additional parameter which is introduced for the sake of analysis First we need a few auxiliary results given in the following lemmas Lemma 61 For all C, D R n n with D symmetric positive definite it holds that C D = C T D Copyright by IAM Unauthorized reproduction of this article is prohibited

14 FOV ANALYI OF AUGMENTED LAGRANGIAN METHOD 783 Proof This simple result can be shown as follows: Cx D D 1 2 Cx,y C D =max = max x R n x D x,y R n x D y = max x, C T D 1 2 y x,y R n x D y x, C T y D 1 2 x, C T y x, D 1 2 C T y = max = max = max x,y R n x D y D x,y R n x y D x,y R n x y D D 1 2 C T y =max = C T y R n y D D We recall that A is the symmetric part of A i, j {1, 2}; then we can write Denote A s ij = 1 2 A ij + A T ij ), ) A s A = 11 A s 12 A s 12 A s 22 Note that A s 11 and As 22 are both symmetric positive definite With this notation and the one introduced in 34), the following lemma holds Lemma 62 For any ν>0, α 0 and any constant κ 0, 1), thereexistsa sufficiently small >0 such that A 11 AT 12 A s κ, 22 As A A 12 A s 11 A s κ, 22 61) A T 11 AT 12 A s κ, 22 As A T A 12 A s 11 A s κ 22 hold independently of the mesh size h Proof We will check the first inequality in 61); the second follows by similar arguments For arbitrary w R n consider u = A 11 AT 12w To prove the inequality A 11 AT 12 A s κ, we need to show that 22 As 11 62) A s 11 u, u κ2 A s 22w, w Consider the FE counterparts of u and w, thatis,u h and w h from V h V h is the FE space for one component of the velocity, so V h = V h V h ) Using the definition of the matrices involved, checking 62) is equivalent to showing that 63) α u h 2 + ν u h 2 + P h u h ) x 2 κ 2 α w h 2 + ν w h 2 + P h w h ) y 2 ) ince A 11 u = A T 12 w, the functions u h and w h satisfy the relation 64) αu h,v h )+ν u h, v h )+P h u h ) x, v h ) x )+a ) u h,v h ) = P h w h ) y, v h ) x ) v h V h Letting u h = v h in 64) and using the Cauchy chwarz inequality and P h 1 since P h is an L 2 orthogonal projector), one verifies 63) for any cν for some mesh- and parameter-independent constant c The arguments for proving the third inequality in 61), ie, A T 11 AT 12 A s 22 As 11 κ, are the same with the only difference being the minus sign in front of a ) u h,v h ) in 64) This sign does not matter, since the term cancels out for the choice of v h = u h The second and fourth inequalities in 61) are proved similarly Consider now the following representations for the preconditioned velocity blocks: ) A Â = I C, with C 0 0 =, A 12 A 11 A 12 A 11 AT 12A 22 Copyright by IAM Unauthorized reproduction of this article is prohibited

15 784 MICHELE BENZI AND MAXIM A OLHANKII and  A = I Ĉ, with Ĉ = A 11 AT 12 A A 22 A A 12 0 ) For the matrices C and Ĉ we prove the following result Lemma 63 For any ν>0, α 0 and any constant κ 0, 1), thereexistsa sufficiently small >0 such that 65) C A κ and Ĉ A κ hold independently of the mesh size h Proof By Lemma 61, the first bound in 65) is equivalent to 66) C T A κ 1 Consider the natural splitting: C T = C 1 + C 2, with C 0 A T 1 = 11 AT Hence for 66) it is sufficient to show that 67) C 1 A 1 2 κ 1 and C 2 A 1 2 κ 1 ) 0 0, C2 = 0 A T 22 A 12A T 11 AT 12 By the definition of A and C 1, the first inequality in 67) is equivalent to A T 11 AT 12 A s 22 As κ 1 The latter holds by Lemma 62 To show the second inequality in 67), we apply Lemma 62 once again through the following decomposition: C 2 A = A T 22 A 12A T 11 AT 12 A s 22 A s 22 A T 22 A 12 A s 11 A s 22 A T 11 AT 12 A s 22 A s 11 With the help of Lemma 62, the bound for Ĉ in 65) is proved by arguments similar to those used above to show 66) For the analysis of the inexact AL preconditioner we apply Theorem 39 of [15] For the reader s convenience we give this result below in the notation of our paper Theorem 64 see [15]) Denote Ŝ = B B T Assume that ) 68) 69) 610) 611) inf z 1 R n+m z 1 R n+m A  μ a inf u R n p u, u A u 2 A A z 1, z 2 C 1, z 2 R z n+m 1 a z 2 a A z 1, z 2 C 2, z 2 R z n+m 1 a z 2 a, A  A Γ a, ŜW q, q M p μ s ν + ) inf q R m q 2, ν + ) ŜW M M p Γ s Copyright by IAM Unauthorized reproduction of this article is prohibited

16 FOV ANALYI OF AUGMENTED LAGRANGIAN METHOD 785 hold with some positive mesh-independent constants C 1,C 2,μ a, Γ a,μ s, Γ s Then there exists ρ 0 > 0 such that for all ρ ρ 0 it holds that 0 <a 1 μa P ), PA a a 2 with some mesh-independent constants a 1, a 2, provided that 612) I A Â A ρ We remark that the proof of this result in [15] gives the following sufficient lower bound on ρ 0 : ρ 0 C 2Γ a + C2 2C 2 1 Γ sμ a )2 + μ 2 s 2μ a μ s The main result of this section is the following theorem Theorem 65 For any ν > 0 and α 0 and sufficiently small > 0 and large enough ρ, the preconditioned GMRE method with modified AL preconditioner satisfies the convergence estimate r k a q k r 0 a, with some h-independent positive constant q<1 Proof We shall use the result of Theorem 64 The estimates 68) and 69) were already shown in 35) and 36) with mesh-independent constants C 1 and C 2 estimated in Lemma 21 Thus, it remains to show 610), 611) with some meshindependent constants μ a, Γ a and μ s, Γ s First we check 610) Based on the splitting A Â = I C and Lemma 63, one finds for sufficiently small >0 A Â A u, u = u 2 A Cu,u A and A Â A 1 C A ) u 2 1 κ) u 2 A A 1 + C A ) 1+κ, with some mesh-independent κ 0, 1) Thus we check that 610) holds with some mesh-independent constants μ a and Γ a Now we turn to proving 611) For the choice of W = M p the bounds in 611) are equivalent to Ŝq, q 613) μ s ν + ) inf q R m q 2, ν + ) Mp Ŝ M Γ s M Thanks to the lower bound for the chur complement of the augmented system in 55), for the first inequality in 613) it is sufficient to show that 614) q, q Ŝ q, q q R m Copyright by IAM Unauthorized reproduction of this article is prohibited

17 786 MICHELE BENZI AND MAXIM A OLHANKII ince = BA BT and Ŝ = B BT, the inequality 614) follows from A u, u  u, u A u, u A u, u 1 A u, u  A u, A u u R n u R n  [ ] [ ] A u, A u [ 2  + ÂT ) A u ] u R n [Â, u] A u R n Nowwenotethatfor ν it holds that A u, u 1 2  + ÂT )u, u for all u Rn Thus, for sufficiently small, estimate 614) follows from [ ] [ ] 615) u, u A A u, A u u R n A To show 615) we apply Lemma 63: [ ] [ ] A u, A u = I A Ĉ)u 2 A 1 Ĉ A ) 2 u 2 A 1 κ) 2 u 2 A, where κ is a mesh-independent constant less than 1 Thus, the first inequality in 611) is proved The second inequality in 611) is proved by the following argument For W = M p we get 616) ŜW M p = M 1 2 p Ŝ M 1 2 p = M 1 2 p B BT M 1 2 p = M 1 2 p BD )D 2  D 1 2 )D 1 2 B T M 1 2 p ) D 1 2 B T M 1 2 p 2 D 1 2  D 1 2 For the first factor on the right-hand side of 616) it holds that 617) D 1 2 B T M 1 D 1 2 B T q 2 p = q R m q M = q R m = q R m q, B u u R n q M D 1 2 u = D 1 2 B T q, u u R n q M u q h Q h q h, div u h ) u h V h q h u h 1 It remains to estimate the second factor on the right-hand side of 616) First, we get by Lemma 61 Next, we compute D 1 2   T D = D 1 2 = D  A T D =  T 11 D 0 A T 22 A 12A T 11 D A T D D 22 D ) Copyright by IAM Unauthorized reproduction of this article is prohibited

18 FOV ANALYI OF AUGMENTED LAGRANGIAN METHOD 787 Note that D is a block diagonal matrix and the D-norm of every block of the matrix  T D can be estimated separately For example, let us consider the 1,1) block A T 11 D By the definition of the matrices involved, the bound A T 11 D D C 1 is equivalent to the estimate 618) u h C 1 w h, with arbitrary w h V h and where u h V h solves 619) αu h,v h )+ν u h, v h )+P h u h ) x, v h ) x ) a ) u h,v h ) = w h, v h ) v h V h etting v h = u h in 619) immediately gives 618) with C 1 = ν 1 2 imilarly, A T 22 D D ν 1 2, and, finally, with the same arguments A T 22 A 12A T 11 D D A T 22 A 12 D A T 11 D D ν This completes the proof 7 Conclusions In this paper we have used field-of-values analysis to establish rigorous convergence bounds for GMRE with various AL-based preconditioners of block triangular type applied to the discrete Oseen problem In particular, we proved that for suitable choices of the parameter the GMRE method with the ideal exact) AL preconditioner converges at a rate independent of both the mesh size h and the viscosity ν For the modified inexact) AL preconditioner, the GMRE rate of convergence is proved to be h-independent Thus, the theory developed in this paper provides a rigorous justification for the convergence behavior observed in practice and reported in the papers [2, 3, 4] REFERENCE [1] M Benzi, G H Golub, and J Liesen, Numerical solution of saddle point problems, Acta Numer, ), pp [2] M Benzi and M A Olshanskii, An augmented Lagrangian-based approach to the Oseen problem, IAM J ci Comput, ), pp [3] M Benzi, M A Olshanskii, and Z Wang, Modified augmented Lagrangian preconditioners for the incompressible Navier tokes equations, Internat J Numer Methods Fluids, to appear; also available as DOI: /fld2267 [4] M Benzi and Z Wang, Analysis of augmented Lagrangian-based preconditioners for the steady incompressible Navier tokes equations, IAM J ci Comput, to appear [5] Berrone, Adaptive discretization of the Navier tokes equations by stabilized finite element methods, Comput Methods Appl Mech Engrg, ), pp [6] P G Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, Amsterdam, 1978 [7] C Eisenstat, H C Elman, and M H chultz, Variational iterative methods for nonsymmetric systems of linear equations, IAM J Numer Anal, ), pp [8] H Elman and D ilvester, Fast nonsymmetric iterations and preconditioning for Navier tokes equations, IAM J ci Comput, ), pp [9] H C Elman, D J ilvester, and A J Wathen, Finite Elements and Fast Iterative olvers: With Applications in Incompressible Fluid Dynamics, Numer Math ci Comput, Oxford University Press, New York, 2005 [10] M Fortin and R Glowinski, Augmented Lagrangian Methods: Applications to the Numerical olution of Boundary-Value Problems, tud Math Appl 15, North Holland, Amsterdam, 1983 [11] T Gelhard, G Lube, M A Olshanskii, and J-H tarcke, tabilized finite element schemes with LBB-stable elements for incompressible flows, J Comput Appl Math, ), pp Copyright by IAM Unauthorized reproduction of this article is prohibited

19 788 MICHELE BENZI AND MAXIM A OLHANKII [12] G H Golub and C Greif, On solving block-structured indefinite linear systems, IAMJ ci Comput, ), pp [13] M D Gunzburger, Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms, Academic Press, Boston, 1989 [14] A Klawonn and G tarke, Block triangular preconditioners for nonsymmetric saddle point problems: Field-of-value analysis, Numer Math, ), pp [15] D Loghin and A J Wathen, Analysis of preconditioners for saddle-point problems, IAM J ci Comput, ), pp [16] M A Olshanskii and M Benzi, An augmented Lagrangian approach to linearized problems in hydrodynamic stability, IAM J ci Comput, ), pp [17] Y aad, Iterative Methods for parse Linear ystems, 2nd ed, IAM, Philadelphia, 2003 [18] M ur Rehman, C Vuik, and G egal, A comparison of preconditioners for incompressible Navier tokes solvers, Internat J Numer Methods Fluids, ), pp [19] A J Wathen, Realistic eigenvalue bounds for the Galerkin mass matrix, IMA J Numer Anal, ), pp Copyright by IAM Unauthorized reproduction of this article is prohibited