Preconditioning techniques for Newton s method for the incomressible Navier Stokes equations H. C. ELMAN 1, D. LOGHIN 2 and A. J. WATHEN 3 1 Deartment of Comuter Science, University of Maryland, College Park, MD 20742, USA, email: elman@cs.umd.edu 2 Parallel Algorithms Project, CERFACS, 42 ave G. Coriolis, Toulouse, 31057, France, email: loghin@cerfacs.fr 3 Oxford University Comuting Laboratory, Parks Road, Oxford, OX1 3QD, email: wathen@comlab.ox.ac.uk CERFACS Technical Reort TR/PA/03/31 Abstract Newton s method for the incomressible Navier Stokes equations gives rise to large sarse non-symmetric indefinite matrices with a so-called saddle-oint structure for which Schur comlement reconditioners have roven to be effective when couled with iterative methods of Krylov tye. In this work we investigate the erformance of two reconditioning techniques introduced originally for the Picard method for which both roved significantly suerior to other aroaches such as the Uzawa method. The first is a block reconditioner which is based on the algebraic structure of the system matrix [4]. The other aroach uses also a block reconditioner which is derived by considering the underlying artial differential oerator matrix [9]. Analysis and numerical comarison of the methods are resented. Keywords: Block reconditioners, Navier Stokes, Newton s method. 1 Problem descrition Mixed finite element discretizations of Navier Stokes equations give rise to nonlinear systems with the following saddle-oint structure (see, e.g.,[7], [14]) ( ( ) ( u F (u) B t u K(u) = = f (1) ) B 0 ) where K(u) R n n, F (u) R n1 n1 is nonsymmetric and ossibly indefinite and B t R n1 n2 has nontrivial kernel sanned by the constant vector. 1
Since K is large and sarse, a cometitive class of solution methods is that of iterative solvers of Krylov tye combined with suitable reconditioning. In this work we are interested in the case where a Newton linearization is emloyed to solve (1); this leads to an iteration of the form [14, 10.3] ( ) ( ) ( ) J(u k δu k+1 F (u ) = k ) + M(u k ) B t δu k+1 B 0 δ k+1 = r k, (2) δ k+1 where M(u) = A u (u) u, u k+1 = u k + δu k+1, r k = f K(u k )u k. On the other hand, the Jacobian matrix J(u k ) is related to the system matrix arising from a Picard linearization of (1), ( ) ( ) ( ) K(u k δu k+1 F (u ) = k ) B t δu k+1 B 0 δ k+1 = r k, (3) δ k+1 for which efficient reconditioners are available. Our aim is to adat and analyze the reconditioned iterative solvers devised for the Picard iteration for roblems where Newton s method is emloyed. Henceforth, we will use the notation F (ρ, u k ) = F (u k ) + ρm(u k ); (4) where ρ {0, 1} to denote the matrices arising from the above two discretizations. To comlete the icture, we note here that for flow roblems in R d (d = 2, 3) there holds F (0, u k ) = I d F (u k ), where F (u k ) is nonsymmetric and ositive-definite and comes from the discretization of an advection-diffusion oerator F (u k ) = νl + N(u k ), where ν is the diffusion arameter and L is a discrete Lalacian. The matrix M(u k ) is also a d d block-matrix and reresents the discretization of a zero-order oerator. Hence it is well-conditioned; however, deending on the roblem, it can be indefinite which leads to the ossibility of F (1, u k ) being indefinite. Finally, we note that the arameter ν is very imortant in ractical alications. Its recirocal, ν, is the Reynolds number and given the size of the roblem n, there are arameters ν for which the nonlinear roblem (1) does not have a unique solution. These critical values of ν where non-uniqueness sets in can be detected in various ways, e.g., continuation methods. For this reason, otimality of our solvers will be measured in terms of deendence of the number of iterations on two arameters: n and ν. Whether Picard, Newton or variations thereof, linearizations of (1) reserve the saddle-oint structure of the system matrix for which Schur comlement reconditioners erform otimally. More recisely, it is known that the key to efficient reconditioning is the Schur comlement S = BF (ρ, u k ) B t ; (5) 2
in articular, a block (right) reconditioner of the form ( ) F (ρ, u P = k ) B t 0 S leads to convergence of any Krylov subsace iterative solver in 2 iterations [13], [8]. The difficulty with this aroach obviously relates to alying the action of the inverse of the Schur comlement (5) as art of the reconditioning technique. It is usually the case that the Schur comlement is not available, due to its dense structure and that suitable aroximations (or factorizations) are required if we are to stand any chance to make this aroach cometitive. Two successful Schur comlement aroximations have been devised, analyzed and tested for the case of the Picard iteration (ρ = 0). We consider these aroaches below. The first attemt to aroximate the Schur comlement corresonding to the Picard iteration with a nonsymmetric matrix was resented by Elman in ([4]) who suggested the aroximation S (BB t ) (BF (u k )B t )(BB t ). It is shown in ([4]) that the erformance of the block reconditioner (6) with this choice of aroximation has a mild deendence on both ν and n for the Picard iteration. We refer to ([4]) for the derivation and details; we note however that this choice of aroximation is algebraic, i.e., it can be generalized to any roblem of saddle-oint tye and is indeendent of the underlying differential oerator corresonding to F (ρ, u k ). For this reason, we will take our first aroximation of the Schur comlement of the Newton method to be S 1 = (BB t ) (BF (1, u k )B t )(BB t ). The other aroach we consider here is derived in ([9]). In this aroach, the Schur comlement is seen as the discretization of the fundamental solution for the differential oerator which gives rise to K(u k ), the Picard linearization matrix. The resulting aroximation of the Schur comlement is also defined as a factorization of sarse matrices S 2 = M F A where M, F, A are rojections onto the ressure sace of the identity, advection-diffusion and Lalace oerators, resectively (see ([9]) for details. The cost of assembling the reconditioner in this fashion is justified by the imroved erformance which becomes linear in the size of the roblem, although it still exhibits a mild deendence on ν. Unlike the first choice above, this reconditioner deends strongly on what the underlying oerator is: if F (ρ, u k ) changes, the new underlying oerator will lead to a new fundamental solution which will have to be comuted (if it exists) and discretized. Since F (1, u k ) corresonds to an oerator with non-constant coefficients, the fundamental solution is not available to our knowledge. For this reason we have not been able to adat this reconditioner to Newton s method. We therefore analyze and test the above Schur comlement aroximation S 2 (derived for the Picard iteration) on Newton s method. (6) 3
2 Eigenvalue analysis In this section we resent an analysis of the above reconditioning techniques for the case where the system matrix results from a mixed finite element discretization of the Navier Stokes equations. These discretizations have the advantage of satisfying certain stability conditions which arise in the formulation of the roblem. More recisely, we assume that the following conditions (known as the Babuška-Brezzi conditions) are satisfied by the Jacobian matrix J(u k ) for all u k and for all ν max max w R n \{0} v R n \{0} min max w R n \{0} v R n \{0} w t J(u k )v v H w H Γ (7a) w t J(u k )v v H w H γ, (7b) where Γ, γ > 0. These conditions guarantee existence and uniqueness of a finite element aroximation ([6, Cha. IV]). Here H is a symmetric ositive-definite matrix which defines the discrete norm of the roblem; this norm deends on the choice of function saces where the solution is sought. For the case of the standard mixed finite-element aroximations of the Navier Stokes equations the norm-matrix H takes the form ( ) A 0 H =, 0 M where A = I d L is a d-dimensional Lalacian matrix and M is a Grammian matrix (mass matrix) assembled on the ressure sace. Let us consider a general block-triangular reconditioner of the form (6) with F (ρ, u k ) and S relaced by the resective aroximations F, Ŝ ( ) F B t P =. (8) 0 Ŝ The reconditioned matrix is then ( F JP ˆF (I F = ˆF )B t Ŝ B ˆF SŜ and it aears that the derivation of eigenvalue bounds for the above system is not a simle task. For this reason we make recourse to the general analysis of block-reconditioners for saddle-oint roblems resented in [11]. We recall here the results of interest. Definition 2.1 H-norm equivalence Non-singular matrices M, N R n n are said to be H-norm equivalent if there exist constants α, β indeendent of n such that for all x R n \ {0} ). We write α Mx H Nx H β. M H N. 4
Thus, H-norm equivalence simly means that the H-singular values of MN are bounded. An immediate consequence is that the eigenvalues of the reconditioned system are bounded indeendently of the size of the roblem α < λ(mn ) < β; moreover, H-norm equivalence is an equivalence relation on R n n. We will need the following results roved in [11]. Lemma 2.2 Let (7) hold. Then Moreover, if for all v R n1 n1 H H J. v t F (1, u k )v η v 2 A (9) then the Schur comlement S = BF (1, u k ) B t satisfies S M M. (10) Lemma 2.3 Let (7) hold with H defined as above and let P be defined in (8). Then P H J if F A A, Ŝ M M. The above results simlify our task considerably; in articular, under standard assumtions on the matrix J, it is sufficient to erform our analysis on the equivalence of the diagonal blocks of H and P. Let us consider first a norm-equivalent aroximation F. Since F (0, u k ) is the discretization of a (ositive-definite) advection-diffusion oerator it satisfies (see, e.g., [15]) v t F (0, u k )v η v 2 A v R n1 n1. (11) This bound imlies that F (0, u k ) satisfies a min-max bound of tye (7b) with resect to the A-norm; since F (0, u k ) also satisfies a max-max bound with resect to the same norm, we can aly Lemma 2.2 to deduce that F (0, u k ) A A. It can be shown that there exists a value ν 0 such that a bound of tye (9) holds also for F (1, u k ) [6,. 300] and thus F (1, u k ) A A. We remark here that the value of ν 0 is not known a riori and that it deends on alication. We will investigate this issue in the numerics section. Remark 2.1 In general, even the weaker bound min max w R n 1 \{0} v R n 1 \{0} w t F (1, u k )v v A w A η (12) is not imlied by the stability conditions (7). In fact, the Babuška-Brezzi conditions imly that the above min-max bound only holds over kerb [1]. 5
Since the inverse of the matrix F (1, u k ) can be quite exensive to comute in ractice we consider below two block aroximations. For this urose, let us look at the structure of F (1, u k ) for two-dimensional roblems (cf. (4)) ( ) ( ) F (1, u k F (u ) = k ) 0 M11 (u 0 F (u k + k ) M 12 (u k ) ) M 21 (u k ) M 22 (u k. ) Simle aroximations are given by ( F T (u k F (u ) = k ) + M 11 (u k ) M 12 (u k ) 0 F (u k ) + M 22 (u k ) ). (13) and F D (u k ) = ( F (u k ) + M 11 (u k ) 0 0 F (u k ) + M 22 (u k ) ). (14) Given their structure, F T (u k ), F D (u k ) satisfy (7a) if F (1, u k ) does. If (9) holds for F (1, u k ) then F D (u k ) satisfies (9) also and thus F D (u k ) A A. (15) If moreover M 12 (u k ) < η, then one can show that v t F T (u k )v 1 2 η v 2 A ; thus F T (u k ) A A. (16) We note that for a quasi-uniform discretization (see below) M 12 (u k ) = O(h 2 ) so that the above restriction is satisfied if η > ch 2. However, η = O(ν) and ν > ν 0 is a stronger restriction on M 12 (u k ). Henceforth, we restrict our attention to the regime ν > ν 0, for which F (1, u k ), F T (u k ), F D (u k ) are all norm equivalent to A. This equivalence is useful in the analysis we resent below. In the following we assume a quasi-uniform discretization of our hysical domain, i.e., we assume that we have a subdivision of our comutational domain in which all the simlices do not exceed a diameter h. Moreover, the following bounds can be shown to hold for all v R n1, q R n2 \ kerb t and Ω R 2 c1h 2 v 2 v t Av c2 v 2, (17a) c 3 h 4 q 2 q t BB t q c 4 h 2 q 2, c 5 h 2 q 2 q t M q c 6 h 2 q 2. (17b) (17c) The following results rovide eigenvalue bounds for the system matrix reconditioned in the fashion described in the revious section. Theorem 2.4 Let (7) hold and let ν > ν 0 be a viscosity arameter such that F (1, u k ) satisfies (9). Let ( ) F (1, u P i = k ) B t (18) 0 S i 6
where S i, i = 1, 2 are defined as above. Then there exist constants C i, i = 1,..., 4 indeendent of n such that C 1 λ(jp 1 ) C 2 h 2 C 3 λ(jp 2 ) C 4. Proof We first rove the bounds for P 2. Since by hyothesis (9) holds, by the above discussion F (1, u k ) A A; in articular, (7a) imlies F (1, u k )A A = F (1, u k )v max A v n 1\0 Av A Γ. (19) According to Lemma 2.3 we only have to check that S 2 M M. This is equivalent to showing that there exist constants α, β such that α AF x M x M β. On the other hand the above inequality holds with resect to the l 2- norm ([12], see also [10] ) and the required result follows from the sectral equivalence between the mass matrix M and the identity [16]. The bound for P 1 follows in a similar way, the only difference being that we have to show that β is of order h 2 in this case. We have M S 1 M = M 1/2 S 1 M 1/2 c 6h 2 (BB t ) BF (1, u k )B t (BB t ) c 6h 2 A /2 F (1, u k )A /2 (BB t ) BAB t (BB t ) c 6Γh 2 (BB t ) BAB t (BB t ) c 6Γh 2 A (BB t ) c 6c 2c 3 Γh 2 where we used (19) and the fact that A /2 F (1, u k )A /2 = F (1, u k )A A. For the lower bound we note that S 1M M = M /2 S 1M /2 1 σ min(m 1/2 S 1 M 1/2 ). Let S 0 = (BB t )(BAB t ) (BB t ) and let n 2 = n 2 \ kerb t. We have σ min(m 1/2 S 1 M 1/2 ) min q n 2 q t M 1/2 S 1 M 1/2 q t q r t S 1 = min r r t S 0 r r n 2 r t S 0 r r t (BA B t ) r z t F (1, u k )z min z n 2 z t Az η 1 c q min r n 2 r t (BA B t ) r r t M r r t S 0 r r t (BA B t ) r r t (BA B t ) r r t M r 7
where we used Lemma (2.2), the fact that F (1, u k ) satisfies (9) and the result roved in ([4]) that the smallest eigenvalue of is greater or equal to 1. (BA B t )x = λs 0x x n 2 Let Pi T, P i D denote reconditioners of tye (18) with F (1, u k ) relaced by F T (u k ), F D (u k ) resectively. The following result shows that the bounds of Theorem 2.4 hold also for these aroximations. Corollary 2.5 Let the conditions of Theorem 2.4 hold and let Pi denote either of Pi T, P i D. Then there exist constants Ci, i = 1,..., 4 indeendent of n such that C1 λ(jp1 ) C2 h 2 C3 λ(jp 2 ) C4. Proof The roof follows similarly by using the norm-equivalences (15), (16). Remark 2.2 The above analysis relies on relation (9). Under this assumtion, the analysis for the reconditioned matrix JP2 is equivalent to that erformed in [10] for reconditioning the Picard iteration. However, the general analysis in [11] allowed us to simlify that analysis and also to rovide for the first time analytic bounds for the eigenvalues of JP1. Remark 2.3 The above result exhibits only the deendence of the eigenvalues on the mesh arameter. Since by hyothesis, the analysis is restricted to a sufficiently large ν, we will not ursue the deendence on ν here. For the Picard iteration, analyses and exeriments on ν deendence are contained in [4], [5], [10]. Numerical exeriments indicate that the bounds on the eigenvalues are tight. However, it is well-known that the eigenvalues alone do not always describe convergence of nonsymmetric iterative solvers. This is also the case here: although the largest modulus eigenvalues of the reconditioned systems behave as described theoretically, they actually belong to a small grou of outliers. We found exerimentally that most of the eigenvalues are clustered in a region of the lane close to 1, with a small number of eigenvalues migrating from the cluster as the arameters ν, h are reduced. A general qualitative model for convergence of GMRES when clustering occurs can be found in [2]. We recall here their result for one cluster, since it aears to be ertinent to the distribution of eigenvalues resulting from our reconditioning techniques. Given a cluster of eigenvalues C = {λ : λ c < ρ c } to which there corresond M = n C outliers, whose maximum distance δ away from C is λ j z δ = max max z c =ρ c 1 j M λ j 8
h(n) maxre(λ) maxim(λ) n o (2, 1) n o (3, 2) n o (4, 3) 1/16(289) 46.65 6.44 15.2 7.6 3.1 1/32(1089) 177.55 24.72 15.4 7.8 3.6 1/64(4096) 694.42 96.74 15.6 7.9 3.8 Table 1: Sectral information for JP 1 ; n o (c, r) is the ercentage of eigenvalues outside the circle of radius r centered at c. the GMRES residual at ste k + d satisfies r k+d Cρ k r 0 where d is the degree of the minimum olynomial associated with the outliers and C ρδ d is indeendent of k. A secific numerical study of this henomenon for reconditioning with P 2 is contained in ([5]). In this case the location of the eigenvalues does not deend on the mesh-size and the migration of eigenvalues from the cluster occurs with reducing ν. This leads to a delay in convergence deendent on the number of eigenvalues outside the cluster. A very similar behaviour is observed for fixed ν in the case of reconditioning with P 1. As described by Thm 2.4, the maximum modulus eigenvalue of the reconditioned system is bounded from above by a factor of order O(h 2 ). This is in fact a descritive bound. Table 1 exhibits this deendence for the case ν = 1/10 for the driven cavity test roblem (see next section). However, convergence is not entirely described by the behaviour of the largest modulus eigenvalues indeed, the sectrum is mostly clustered in a small region of the comlex lane with only a handful of outliers exhibiting the O(h 2 ) behaviour. Thus, there aears to occur a clustering similar to the case described above where we recondition with P 2 and where convergence is described through a delay deendent on the (increasing) number of outliers. Table 1 resents also the ercentage of (outlying) eigenvalues n o (c, r) outside the circle of radius r centered at c. We see indeed that the number of eigenvalues outside each disc close to the unity, though in small roortion, is increasing with n. This aears to match the behaviour exhibited in the tests below, where a slight deendence on the size of the roblem is noticed, though never a linear one. 3 Numerical exeriments In this section we resent numerical exeriments obtained for a standard test roblem, the driven cavity flow. This involves solving the Navier Stokes equations inside the unit square with Dirichlet boundary conditions equal to zero everywhere, excet for the boundary y = 1 where the horizontal velocity has a ositive rescribed rofile. Although our analysis considered the exact Newton s method, in ractice it is of considerable advantage to use an inexact version for reasons of comutational efficiency. The algorithm we use is a Newton-GMRES method which involves solving (2) using GMRES with reconditioning and 9
the following stoing criterion suggested in [3]: writing (1) as F(w) = 0, where w = (u, ), at each Newton ste i we sto after k iterations of GMRES if the residual r k satisfies r k / F(w i ) c F(w i ) q, (20) where the choice c = 10 2, q = 1/4 does not affect the number of nonlinear iterations in our tests. Moreover, the order of convergence is only marginally affected, as we show below. We reort below the erformance of reconditioners P i, Pi T, P i D, i = 1, 2. The tests were erformed on discretizations resulting from three successively refined meshes and a range of diffusion arameters ν. The value ν 0 in Thm 2.4 deends on the roblem; for our examle we found exerimentally that ν 0 1/80. As the results below demonstrate, the regime ν ν 0 is also the regime where our reconditioners erform best. The finite element discretization is the so-called Q2 Q1 discretization, using a regular mesh of rectangles with finite element bases of quadratic and linear iecewise olynomials for the velocity and ressure, resectively. The initial guess for the Newton iteration was the zero vector, excet for the range ν 1/640 for which three stes of the Picard iteration were necessary to rovide a initial iterate which ensured convergence of Newton s method. We note that this will lead to an aarent imrovement of erformance in this regime, although not to a qualitative change. We stoed the Newton (outer) iteration when F(w i ) / F(w 0 ) 10 6. The comutational cost of our reconditioners is dominated by the inversion of the (1,1)-block F (1, u k ). For the Q2 Q1 discretization the cost of storing the Schur comlement aroximations S i is aroximately 25n 2 (BB t ) +45n 2 (BF B t ) for S 1 and 3 9n 2 (A, F, M ) for S 2. The cost of alying these reconditioners will of course deend on the choice of imlementation. In the exeriments below we imlemented them exactly; however, iterative techniques are certainly ossible [4], [9] in which case the choice S 2 becomes clearly more cometitive. The results for the exact case, when reconditioners P 1, P 2 are emloyed are shown in Tables 2, 3. We see indeed that while the erformance of P 2 is mesh-indeendent, the erformance of P 1 exhibits a sub-linear deendence on the size of the roblem. This we believe to be exlained by the ν = 1/10 1/20 1/40 1/80 1/160 1/320 1/640 1/1280 n = 2, 467 13.7 14.2 15.5 17.7 21.4 29.3 31.8 43.2 9, 539 20.2 21.2 21.5 23.5 29.2 36.8 40.0 53.2 37, 507 29.7 31.7 32.7 33.7 41.2 52.0 62.6 72.3 Table 2: Average number of GMRES iterations er Newton ste for P 1. ν = 1/10 1/20 1/40 1/80 1/160 1/320 1/640 1/1280 n = 2, 467 13.5 13 17.7 22.8 27.8 47.3 59.5 85.1 9, 539 11.5 13.2 16.2 21.2 31.6 44.6 57.2 77.3 37, 507 11.7 14.2 16.5 19.5 29.6 43.5 56.5 75.1 Table 3: Average number of GMRES iterations er Newton ste for P 2. 10
ν = 1/10 1/20 1/40 1/80 1/160 1/320 1/640 1/1280 n = 2, 467 13.7 14.2 16.2 19.5 26.0 32.6 37.6 49.1 9, 539 20.2 22 22.2 25.7 43.2 44.3 44.5 59.7 37, 507 30.2 32 34.5 37.7 49.2 64.0 69.8 80.5 Table 4: Average number of GMRES iterations er Newton ste for P T 1. ν = 1/10 1/20 1/40 1/80 1/160 1/320 1/640 1/1280 n = 2, 467 14.7 14 19.7 26.8 36.4 58.3 71.3 103.5 9, 539 12.5 14.7 17.7 24.7 37.4 55.8 67.2 102.6 37, 507 12.7 15.5 18.2 21.7 34.6 53.2 67.5 108.6 Table 5: Average number of GMRES iterations er Newton ste for P T 2. distribution of eigenvalues which was exhibited in Table 1, which showed an increasing number of outliers with decreasing mesh-size. As for the viscosity arameter ν, reconditioner P 1 aears to exhibit a milder deendence than P 2, which makes it comarable to P 1 for high-reynolds number flows, desite its mesh-deendence. The same behaviour is exhibited by the reconditioners Pi T, P i D, i = 1, 2. In articular, the erformance remains mesh-indeendent in the case of P2 T, P2 D as shown in Tables 5, 7; as a consequence, the number of extra iterations comared to the exact case dislayed in Table 3 is indeendent of the mesh. This is not the case for P1 T, P 1 D as shown in Tables 4, 6, for which this difference (cf. Table 2) grows with the mesh arameter, although the number of iterations still grows sub-linearly with resect to the size of the roblem. From a ractical oint of view it is imortant to study the deterioration in erformance when the exact reconditioners P i are relaced with Pi T, P i D, as solving systems with F can be quite costly. As the above numerics show, this deterioration is negligible for large ν and quite accetable for smaller ν. To exemlify this we relaced the exact solution of systems ν = 1/10 1/20 1/40 1/80 1/160 1/320 1/640 1/1280 n = 2, 467 14 15 17 21.5 28.4 36.2 39.2 51.4 9, 539 20.7 23.2 24.5 28.5 37.6 46.8 45.3 59.2 37, 507 30.7 33.7 36.5 41.2 54.0 69.2 69.5 80.83 Table 6: Average number of GMRES iterations er Newton ste for P D 1. ν = 1/10 1/20 1/40 1/80 1/160 1/320 1/640 1/1280 n = 2, 467 15.7 15.7 22.7 30.6 41.6 65.6 77.9 114.3 9, 539 13 16 19.7 25.2 43 63 72.3 111.0 37, 507 13.7 17 20.7 25 39.6 59.2 72.5 116.1 Table 7: Average number of GMRES iterations er Newton ste for P D 2. 11
ν = 1/10 1/20 1/40 1/80 1/160 1/320 1/640 1/1280 F T 3.0 3.8 4.5 5.6 8.1 12.0 16.4 27.7 F D 4.8 6.2 8.3 11.3 15.6 22.5 31.7 34.2 Table 8: Average number of iterations er outer-gmres ste for inner- GMRES solver using reconditioners F T, F D. involving F (1, u k ) with a GMRES iteration with reconditioners F T, F D. The stoing criterion was the reduction of the relative 2-norm residual by a factor of 10 6 which insured that this inner iteration did not affect the outer iteration count dislayed in Tables 2, 3. The number of iterations was always mesh-indeendent. We resent these results in Table 8. We see indeed that the modified reconditioners Pi T, P i D (which require just one solve with F T and F D resectively) outerform by far P i when the inner-gmres iteration is used to aroximate F. Thus, unless a fast solution algorithm is available for F (1, u k ), the ideal erformance resented in Tables 2, 3 may rove too exensive to achieve. We end this section with a note on the convergence rate of Newton s method observed for our inexact Newton-GMRES algorithm. The choice of this algorithm and in articular of stoing criterion (20) was justified by both ractical and theoretical considerations [3]. In Table 9 we dislay the average convergence rate observed in our tests. More recisely, we tabulated the mean of the ratio q k of logs of consecutive residuals q k = log F(wk ) log F(w k ), for the cases q = 1/4 and q = 1 in (20) and for the exact case (q = ). The choice of reconditioner does not affect the convergence rate dislayed below. We see indeed that the choice of stoing criterion (i.e., q in (20)) does not affect significantly the rate of convergence of the method excet ossibly for the low Reynolds number (large ν) regime. In rincile, one could recover this convergence rate by a stricter criterion (q = 1, say), at the cost of more iterations. Given our choice of tolerance for the outer nonlinear iteration (10 6 ), q = 1/4 was sufficient to kee the same number of Newton stes as in the exact case. A stricter tolerance may require a larger value of q. However, for high Reynolds number flows a more economical stoing criterion is sufficient due to the sub-otimal convergence of Newton s method. ν = 1/10 1/20 1/40 1/80 1/160 1/320 q = 1/4 1.78 1.77 1.70 1.58 1.43 1.28 q = 1 2.00 1.98 1.80 1.57 1.43 1.28 q = 2.20 2.02 1.81 1.57 1.43 1.28 Table 9: Convergence rate of Newton s method: exact and inexact with q = 1/4 and q = 1; n = 9, 539. 12
4 Summary The urose of this work was to analyze and comare block reconditioners for the system matrix arising from a Newton iteration for the Navier Stokes equations. The reconditioners are saddle-oint reconditioners using an algebraic and oerator based aroximation of the Schur comlement. They were introduced originally for the case of the Picard iteration. In each case we give analytic bounds for the sectrum of the reconditioned system under standard stability assumtions for the finite element discretization. The bounds require a further assumtion of ositivedefiniteness of the (1, 1)-block of the matrix, which holds for large viscosity arameters. The numerical exeriments however show no qualitative change even for smaller values of viscosity. In each case the imlementation is modular we only require the action of inverses of sub-blocks which have to be constructed algebraically or assembled together with the system matrix. The erformance is very similar to the case of the Picard iteration which has been recently reorted. The oerator-based reconditioner aears to be more robust and less costly than the algebraic reconditioner and esecially the indeendence of the size of the roblem is an attractive feature. However, otimality with resect to the Reynolds number is still to be achieved. References [1] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Sringer-Verlag, New York, 1991. [2] S. L. Cambell, I. C. F. Isen, C. T. Kelley, and C. D. Meyer, GMRES and the minimal olynomial, BIT, 36 (1996),. 664 675. [3] R. S. Dembo, S. C. Eisenstat, and T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal., 19 (1982),. 400 408. [4] H. C. Elman, Preconditioning for the steady-state Navier-Stokes equations with low viscosity, SIAM J. Sci. Com., 20 (1999),. 1299 1316. [5] H. C. Elman, D. J. Silvester, and A. J. Wathen, Performance and analysis of saddle oint reconditioners for the discrete steadystate Navier-Stokes equations, Numer. Math., 90 (2001),. 665 688. [6] V. Girault and P. A. Raviart, Finite Element Methods for Navier Stokes Equations, Sringer-Verlag, 1986. [7] P. M. Gresho and R. L. Sani, Incomressible flow and the finite element method: isothermal laminar flow, Wiley, Chichester, 1998. [8] I. C. F. Isen, A note on reconditioning non-symmetric matrices, SIAM J. Sci. Comut., 23(2001),. 1050 1051. [9] D. Kay, D. Loghin, and A. J. Wathen, A reconditioner for the Steady-State Navier-Stokes equations, SIAM J. Sci. Comut., 24 (2002),. 237 256. 13
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