Block preconditioners for saddle point systems arising from liquid crystal directors modeling

Transcription

1 Noname manuscript No. (will be inserted by the editor) Block preconditioners for saddle point systems arising from liquid crystal directors modeling Fatemeh Panjeh Ali Beik Michele Benzi Received: date / Accepted: date Abstract We analyze two types of block preconditioners for a class of saddle point problems arising from the modeling of liquid crystal directors using finite elements. Spectral properties of the preconditioned matrices are investigated, and numerical experiments are performed to assess the behavior of preconditioned iterations using both exact and inexact versions of the preconditioners. Keywords liquid crystals saddle point problems Krylov methods preconditioning Mathematics Subject Classification (2) MSC 65F 65F8 65F5 Introduction In this paper we study the performance of two types of block preconditioners which can be used in conjunction with Krylov subspace methods for solving large and sparse saddle point systems of the form A B C x A u B C y z = b b 2 b 3 b, () where A R n n and R p p are both symmetric positive definite (SP), B R m n, and C R p n. Sequences of linear systems of the type () appear in liquid crystal modeling; see section 2 and [6]. his paper is a continuation of the recent work [3], where we considered the solution of linear systems of the form () under the hypothesis that is positive semidefinite, including the case =. he remainder of the paper is organized as follows. In section 2 we provide some information on the numerical problem and its origin. Section 3 briefly discusses related work F. P. A. Beik epartment of Mathematics, Vali-e-Asr University of Rafsanjan, PO Box 58, Rafsanjan, Iran f.beik@vru.ac.ir M. Benzi epartment of Mathematics and Computer Science, Emory University, Atlanta, Georgia 3322, USA. Current address: Scuola Normale Superiore, Piazza dei Cavalieri, 7, 5626 Pisa, Italy michele.benzi@sns.it

2 2 Fatemeh Panjeh Ali Beik, Michele Benzi and contrasts the results of this paper with the ones in [3]. In section 4 we describe the two preconditioners and analyze their theoretical properties. Numerical experiments showing the effectiveness of the preconditioners are discussed in section 5, and brief conclusive remarks are given in section 6. Notation. hroughout the paper, for a given square matrix A with real eigenvalues, the notations λ min (A) and λ max (A) denote the minimum and maximum eigenvalues of A, respectively. he set of all eigenvalues (spectrum) of A is represented by σ(a). he symbol A (A ) means that A is a SP (symmetric positive semi-definite) matrix. Moreover for two given matrices A and B, by A B (A B) we mean A B (A B ). he notations ker(b) and range(b) stand for the null space (or kernel) and the column space (or range) of the matrix B, respectively. Given a complex vector x, its conjugate transpose is denoted by x. Moreover, we write (x;y;z) to denote the vector (x,y,z ). 2 he problem As discussed in [6] (see also [, 2]), mathematical models for understanding the orientational properties of liquid crystals lead to the minimization of the free energy, which is a functional of the form F[u,v,w,U] = [ (u 2 2 z + v 2 z + w 2 z) α 2 (β + w 2 )Uz 2 ] dz, (2) where u,v,w and U are functions of the space variable z [,] that are required to satisfy suitable end-point conditions, u z = du dz (etc.), and α,β are prescribed (positive) parameters. he problem is discretized by means of a uniform piecewise-linear finite element method. With a mesh-size h = k+ (with k+ the number of cells), using nodal quadrature and the prescribed boundary conditions leads to replacing the functional F with a function f of 4k variables: where F[u,v,w,U] f(u,...,u k,v,...,v k,w,...,w k,u,...,u k ), f = h k ( ) u j+ u 2 ( ) j v j+ v 2 ( j w j+ w j j= h h h ( )] w 2 α [β 2 j + w 2 (Uj+ ) j+ U 2 j +, 2 h with z j = jh, j =,...,k+, and u j u(z j ), etc. he irichlet end point conditions u() =, v() =, w() =, U() =, u() =, v() =, w() =, U() = are used to eliminate u, v, w, U and u k+, v k+, and U k+. he free energy functional (2) must be minimized subject to the unit vector constraint. For the discrete (finite-dimensional) problem, this constraint is enforced by imposing that the solution components u j, v j and w j are such that ) 2 u 2 j + v 2 j + w 2 j =, j =,...,k.

3 Block preconditioners for a class of saddle point systems 3 Introducing Lagrange multipliers λ,...,λ k, the problem reduces to finding the critical points of the Lagrangian function L = f + 2 k j= λ j (u 2 j + v2 j + w2 j ). Setting the partial derivatives of L to zero results in the nonlinear system of 5k equations in as many unknowns L(x) = where the unknown vector x R 5k contains the values (u j,v j,w j ) with j k, λ j with j k, and U j with j k (in this order). his nonlinear system is solved by Newton s method, leading to a linear system of the form 2 L(x (l) )δx (l) = L(x (l) ) (3) at each step l. Here we denote by 2 L(x (l) ) the Hessian of L evaluated at the current approximate solution x (l). In this paper we investigate efficient preconditioned Krylov subspace methods for the solution of these linear systems, continuing the work in [3]. For issues related to the nonlinear (Newton) iteration, the reader is referred to [6]. As shown in [6], the Hessian has the following block 3 3 structure: A B C 2 L = B, C where A is n n, B is m n, C is p n and is p p with n = 3k and m = p = k. Hence, it is necessary to solve a system of the form () at each Newton step. All the nonzero blocks in A are sparse and structured. A detailed description of the submatrices A, B, C and is given in [6]. Here we mention that A is SP, B has full column rank and is such that BB is diagonal (and indeed BB = I m if the unit vector constraints are satisfied exactly), C is rank deficient, and is tridiagonal and SP. Following [6], for most of our experiments we used the following values of the parameters α and β appearing in (2): α =.5α c and β =.5 where α c 2.72 is known as the critical switching value. In order to assess the robustness of our preconditioners with respect to α, the main control parameter, we also perform some tests with different values of α, see section 5. he following proposition provides a necessary and sufficient condition for the invertibility of A in (). Proposition Assume that A and are both SP matrices. hen matrix A is invertible if and only if B has full column rank. Proof See [3, Proposition 2.]. Hence, the linear systems () arising from the application of Newton s method to the nonlinear system L(x) = are uniquely solvable. We are assuming here that the Hessian is being evaluated away from bifurcation points and turning points, see again [6].

4 4 Fatemeh Panjeh Ali Beik, Michele Benzi 3 Related work A few iterative methods for the solution of saddle point problems arising from liquid crystal director modeling have been proposed in the literature. In [], a coupled multigrid method with Vanka-type relaxation has been developed for solving the 3 3 block-structured linear systems arising at each Newton step. Numerical experiments show very good scalability with respect to problem size for a 2 model problem. For the two largest problem sizes tested, the propsoed multigrid solver outperforms a state-of-the-art sparse direct solver. In [6], a preconditioned nullspace method has been proposed and tested on the same model problems considered in the present paper. he paper also includes tests with MIN- RES used with a block diagonal preconditioner. Although these preconditioners display mesh-independent convergence rates, the timings reported in [6] (see able 6.6) are rather high, given the problem sizes considered (for example, 25 seconds for a problem with k = 65,536), even allowing for the difference in computers used for the tests. his may be due to the high set-up costs incurred by the AMG solver used for the inner solves needed to implement the preconditioners. In our recent work [3] we investigated a variety of block diagonal and block triangular preconditioners for the solution of (), both with = and. For the case, the best results were obtained using inexact variants of the following two block triangular preconditioners: A B C P = BA B, (4) (+CA C ) and A B C Pˆ = BA B BA C. (5) (+CA C ) he theory developed in [3] shows that the spectrum of the exactly preconditioned matrix P ˆ A lies in the interval ( 2,], independent of problem parameters. Numerical experiments demonstrate mesh-independent convergence rates for both preconditioners, and excellent scalability in terms of CPU times when inexact variants are used in combination with FGMRES. In the following section we propose and analyze new block preconditioners for the solution of linear systems of the form () with. hese preconditioners are quite different from those proposed in [3], and overall appear to be superior to the best methods in [3] in terms of CPU times, especially for large problem sizes. 4 escription and analysis of the preconditioners In this section we consider the following two block preconditioners: A B A B C P γb and P γb, (6) where γ is a given real nonzero parameter. he first preconditioner is block diagonal, the second block triangular. Moreover, P is symmetric indefinite for γ =. Note that both matrices are nonsingular under the assumptions of Proposition.

5 Block preconditioners for a class of saddle point systems 5 In order to apply block preconditioners of the form (6) within a Krylov-type method, it is necessary to solve (exactly or inexactly, see below) the following saddle point problem at each step: [ A B γb ][ w w 2 ] = [ r r 2 ]. (7) his system is uniquely solvable for all γ under our assumptions on A and B (see, e.g., [4]). o this end, the following two steps are performed: Step I. Solve Sw 2 = BA r γ r 2 to find w 2, Step II. Solve Aw = r B w 2 to find w, where we have set S = BA B. We defer to section 5 a brief discussion of how the special properties of the matrices A and B arising from the liquid crystal director model may be exploited to perform steps I-II efficiently. he remainder of this section is divided into two main parts. In the first part we present some results on the eigenvalue distribution of the preconditioned matrices corresponding to P and P. he second part is devoted to discussing sufficient conditions which guarantee mesh-independent convergence rates. 4. Spectral analysis of the preconditioned matrices In the following we determine bounds for the eigenvalues of the preconditioned matrices corresponding to the block preconditioners P and P. o this end we establish two theorems. he results are then illustrated numerically on a linear system of moderate size. heorem Suppose that A, and B has full column rank. hen σ(p {, A ) } { } λ max (C ± iµ < µ C). (8) γ λ min (A) Proof Let λ be an arbitrary eigenvalue of P A with the corresponding eigenvector (x;y;z), i.e., Ax+B y+c z = λ(ax+b y), (9) Bx = γλbx, () Cx z = λz. () We claim that λ = is an eigenvalue of P A. Indeed, consider the following two cases: If γ = then λ = is an eigenvalue for P A with corresponding eigenvectors of the form (;y;) where y is an arbitrary nonzero vector. Moreover in the case that ker(c) {}, then (x;y;) is an eigenvector for λ = where x ker(c) and y is an arbitrary vector. If γ then the eigenvectors associated with λ = are of the form (;y;), where y is an arbitrary nonzero vector. Moreover, in the case that ker(b) ker(c) {}, then (x;y;) is an eigenvector corresponding to λ = where x ker(b) ker(c) and y is an arbitrary vector.

6 6 Fatemeh Panjeh Ali Beik, Michele Benzi Now let us focus on the case λ, γ. he assumption λ γ together with () implies that Bx =. Since λ, the vector x must be nonzero, otherwise x = implies that y and z are both zero which is impossible due to the fact that (x;y;z) is an eigenvector. From (9), we derive (λ )x Ax = x C z, (2) and from () it can be deduced that z = λ Cx. Substituting the last expression into (2) we obtain (λ ) 2 x Ax = x C Cx, (3) showing that λ must be purely imaginary, i.e., that λ = ± iµ for some real number µ. he bounds on µ follow from the well-known Rayleigh Ritz heorem, see [5, page 76]. Remark In heorem, assume that γ. From (8) and the proof of the theorem one can see that λ = γ may or may not be an eigenvalue of P A. It can be seen that λ = γ is an eigenvalue if ker(c) range(a B ) {}. (4) he above condition is a sufficient condition. More precisely, if x ker(c) range(a B ), then x = A B y for some nonzero vector y. herefore, we may conclude that λ = γ σ(p A ) with the corresponding eigenvector (x; y;). Conversely, assume that γ σ(p A ). hen the corresponding eigenvectors (x;y;z) must satisfy the following two conditions: ( ) γ [Ax+B C z = y ], z = γ ( γ γ ) Cx. he above conditions are necessary and sufficient for λ = γ with corresponding eigenvectors of the form (x;y;z). to be an eigenvalue of P Next, we consider the eigenvalue distribution for the block triangular preconditioner. o this end, we establish the following theorem. heorem 2 Suppose that A, and B has full column rank. hen all of the eigenvalues of P A are real and { } [ σ(p A ),+ λ max(ca C ] ). (5) γ λ min () Proof Suppose that λ is an arbitrary eigenvalue of P A with corresponding eigenvector (x;y;z), i.e., Ax+B y+c z = λ(ax+b y+c z), (6) Bx = γλbx, (7) Cx z = λz. (8) It turns out that λ = is an eigenvalue of P A. o characterize the corresponding eigenvectors we consider the following two cases: A

7 Block preconditioners for a class of saddle point systems 7 When γ =, then λ = is an eigenvalue with the corresponding eigenvector (x;y;z) for any x ker(c), with the vectors y and z being arbitrary (with at least one of y or z nonzero in case x = ). For the case that γ, the associated eigenvector for λ = is of the form (x;y;z) where x ker(b) ker(c) and y and z are arbitrary vectors. Again, if x = (for example if ker(b) ker(c) = {}), then either y or z must be nonzero. In the rest of the proof we assume that λ, γ. herefore, from (7), we deduce that Bx =. Note that z cannot be zero, otherwise from (6), the positive definiteness of A and the fact that Bx = we would have that x =. But when x and z are both zero vectors, then y = in view of the fact that B has full column rank, which is contrary to our assumption that (x;y;z) is an eigenvector. Consequently, we may assume that z is a nonzero vector such that z 2 =. Using (6), we have y = S BA C z, where S = BA B (note that S is SP, therefore invertible). Multiplying both sides of (6) by CA on the left and using the previous relation, we obtain Cx = CA B S BA C z CA C z. Substituting the above relation into (8), we derive As a result, ( λ)z = CA B S BA C z CA C z. λ = z CA B S BA C z z CA C z z z ( ) z CA 2 I A 2 B S BA 2 A 2 C z = + z. (9) z his expression shows that the spectrum is real. Notice that z / ker(c ), since here z ker(c ) implies λ = which is contrary to our earlier assumption that λ. herefore, from (9) and using the fact that P = I A 2 B S BA 2 is an orthogonal projector and thus P, we conclude that Now the result follows from the Rayleigh-Ritz heorem. < λ + z CA C z z. (2) z Remark 2 Let us consider the case that γ. From (5) it is seen that λ = γ may (or may not) be an eigenvalue of P A. Similarly to Remark, we can see that (4) is a sufficient condition for λ = γ to be an eigenvalue with the same eigenvector structure as given in the previous remark. Moreover, it is easy to see that ( γ,(x;y;z)) is an eigenpair of P A if and only if the following two conditions hold: for γ and (x;y;z) (;;). = Ax+B y+c z, ( ) γ z = Cx γ

8 8 Fatemeh Panjeh Ali Beik, Michele Benzi Fig. op: spectrum of P A for γ = (left), γ = (center) and γ = 2 (right). P A for γ = (left), γ = (center) and γ = 2 (right). Bottom: spectrum of he spectra of each of the preconditioned matrices P A and P A for different values of γ are depicted in Figure for a problem of size hese plots show strong clustering of the eigenvalues, and spectral distributions consistent with those predicted by heorems and 2. We mention that the spectrum of A lies in the union of the intervals [ , ] and [2.56 2, ], for a spectral condition number close to We remark that under a certain additional assumption, we can determine more explicit bounds on the eigenvalues of the preconditioned matrices. o this end, we first recall the following theorem. Here and in the sequel, for a given square matrix A, we denote by ρ(a) the spectral radius of A. heorem 3 [5, heorem 7.7.3] Let A and B be two n n real symmetric matrices such that A is positive definite and B is positive semidefinite. hen A B if and only if ρ(a B), and A B if and only if ρ(a B) <. For the linear systems of the form () arising from liquid crystal modeling, using heorem 3, we could verify numerically that the condition A C C (or equivalently CA C ) holds true for problems of small or moderate size, and it i slikely that the condition holds for larger problems as well. Consequently, in view of (3) and (2), we conjecture that (8) and (5) can be respectively replaced by and σ(p A ) {, γ σ(p A ) } { ± iµ < µ { } [,+ρ( CA C ) ] γ } ρ(a C C) <, { } [,2). γ Indeed, if A C C (or CA C ), then ρ(a C C) = ρ( CA C ) <. 4.2 Further discussion of the properties of the preconditioned matrices In this section we investigate numerically some additional properties of the preconditioned matrices that can be used to shed light on the behavior of the preconditioned iterations.

9 Block preconditioners for a class of saddle point systems 9 Size Extreme eigenvalues of H P S P 2 λ min (H P ) λ max (H P ) able Properties of preconditioned matrices corresponding to P with γ = (left preconditioning). Size Extreme eigenvalues of H P S P 2 λ min (H P ) λ max (H P ) able 2 Properties of preconditioned matrices corresponding to P with γ = 2 (right preconditioning). In the following, we denote the symmetric and skew-symmetric parts of a preconditioned matrix (with either left or right preconditioning) by H P and S P, respectively. It is known that a Krylov subspace method like preconditioned GMRES will exhibit meshindependence convergence rates if the spectra of the matrices H P are entirely contained in an interval [a,b] with < a < b < and if S P 2 c < with a,b, and c independent of the mesh size; see, for example, [8, page 8] and references therein. We have computed the smallest and largest eigenvalues of the symmetric parts of the preconditioned matrices P A, A P, P A and A P for different values of h = k+ and different γ. We have also computed the 2-norm (spectral radius) of the skew-symmetric parts of these same matrices. In the cases corresponding to the block diagonal preconditioner P, the symmetric part of the preconditioned matrices was found to be indefinite. Hence, nothing can be concluded a priori on the behavior of preconditioned GMRES with this preconditioner. For the block triangular preconditioner P, on the other hand, we found that all the quantities of interest appear to be uniformly bounded in the case of left preconditioning with γ = ; see the results displayed in able. In the case of right preconditioning with γ = 2 we have observed a similar behavior, as shown in able 2 (even though in this case there is a very slight dependence on h, which may perhaps disappear for larger problems size). Finally, in Figure 2 we display the spectra of the symmetric parts of the preconditioned matrices for different values of γ using left and right preconditioning with P, with problem size Note that using γ = and right preconditioning leads to an indefinite preconditioned matrix. We emphasize that the above stated conditions for h-independent convergence of preconditioned GMRES are only sufficient, and not necessary. As we will see in the next section, fast, mesh-independence convergence can also be observed in cases where these conditions do not hold.

10 Fatemeh Panjeh Ali Beik, Michele Benzi Fig. 2 Spectra of H P corresponding to P for γ = (left), γ = (center) and γ = 2 (right). op: Left preconditioning. Bottom: Right preconditioning (problem size 2555). 5 Numerical experiments In this section we present the results of numerical experiments aimed at evaluating the convergence behavior of the preconditioned GMRES (PGMRES) and FGMRES methods [7] using preconditioners of the form (6). he computations were performed on a 64-bit 2.45 GHz core i7 processor with 8.GB RAM using MALAB version Right-hand sides corresponding to random solution vectors were used in all of the experiments, performing ten runs and then averaging the CPU-times and rounding iteration numbers to the nearest integer. CPU times and iteration counts are reported under CPU and Iter in the tables below. In the case of FGMRES two values are reported under Iters, namely, the number of steps for the FGMRES method and in parenthesis the total number of inner preconditioned conjugate gradient (PCG) iterations performed over all FGMRES steps. No restarting is used. In all the experiments, the initial guess was taken to be the zero vector and the (outer) iterations were stopped once b A w (k) 2 < 2 b 2, where 2 stands for the Euclidean norm and w (k) = (x (k) ;y (k) ;z (k) ) denotes the current iterate. With this stopping criterion we observe normwise relative errors in the computed solutions of order 5 or smaller. Notice that both the exact and the inexact implementations of the preconditioners require the solution of SP linear systems as subtasks. hese linear systems involve the coefficient matrices A, S = BA B, and. As already mentioned is tridiagonal in our application of interest, therefore the solution of linear systems involving is not an issue. Linear systems with A are also easy to solve, since it turns out that the Cholesky factorization of A (with the original ordering) does not incur any fill-in. Hence, we compute the Cholesky factorization A = LL at the outset, and then perform back and forward substitutions each time the action of A on a vector is required. he only potentially problematic step in applying the preconditioners is the solution of linear systems with the Schur complement S = BA B as the coefficient matrix. his matrix is full and cannot be formed explicitly. Hence, we use the PCG method, where matrix-vector products with S are performed by multiplications with the sparse matrices B and B and solving two bidiagonal linear systems involving L and L. As concerns the choice of preconditioner used for solving linear systems involving the Schur complement S = BA B, we exploit the fact that in the application considered in this

11 Block preconditioners for a class of saddle point systems paper, the matrix B has (nearly) orthogonal rows. his suggests that the matrix BAB might be a good approximate inverse of BA B. he results in able 3 confirm that the eigenvalues of the matrix BAB BA B are bounded and clustered nicely around one, independent of mesh size. Furthermore, it turns out that BAB is actually a tridiagonal matrix, hence the cost of forming, storing, and applying the preconditioner is almost negligible. In Figure 3 we present mesh plots showing the structure of the matrices BAB and (BA B ) (top) and (BAB ) and BA B (bottom). Here the problem size is 2555 and the Schur complement BA B is 5 5. It is clear from these plots that BAB is indeed very close to (BA B ) ; while the latter matrix is full, its off-diagonal entries decay so fast as to make it close to tridiagonal. It is also clear that (BAB ) is quite close to BA B. Fig. 3 Mesh plots of different matrices for the case m = p = k = 5, α =.5α c (where α c = 2.72) and β =.5. op: BAB (left), (BA B ) (right). Bottom: (BAB ) (left), BA B (right). When the block preconditioners are used with GMRES, they must be applied exactly ; i.e., all the linear systems arising as subtasks must be solved to high accuracy. For the systems involving S, where PCG is used, the inner iterations are terminated after the relative residual 2-norm has been reeduced below 2. In contrast, when they are used with FGM- RES, the block preconditioners can be applied inexactly. In this case we use a stopping tolerance of 3 for inner PCG iterations applied to systems with matrix S. Finally, we present iteration counts and CPU times for the various preconditioner and solver options. he results for GMRES preconditioned (on the right) with P are reported

12 2 Fatemeh Panjeh Ali Beik, Michele Benzi Extreme eigenvalues Problem size (m,n) λ min (BAB BA B ) λ max (BAB BA B ) 55 (3,93) (63,89) (27,38) (255,765) (5,533) (23,369) (247,64).3943 able 3 Eigenvalue information for the matrix BAB BA B. Size γ = γ = γ = 2 Iter CPU Iter CPU Iter CPU able 4 Numerical results for PGMRES with the preconditioner P for different γ (right preconditioning). in able 4. As can be seen, convergence is fast and the number of iterations even decreases as the mesh size h = k+ is reduced. he best results are obtained for γ =. Results for FGMRES with the preconditioner P applied inexactly are reported in able 5. Note that for γ = there is no significant deterioration of the rate of convergence when using the inexact preconditioner. he optimality of the PCG iterations used for the inner subtasks is shown by the fact that the total number of PCG steps decreases and appears to be settling around a constant number as h is decreased. As a result, the preconditioned FGMRES solver scales very nicely in terms of CPU time. Also, for large enough problems it is several times faster than the exactly preconditioned GMRES method. Very similar results are obtained with the block triangular preconditioner P. In ables 6 and 7 we report the results for GMRES with the exact P preconditioner for right and left preconditioning, respectively. he latter results are included to show that no significant difference is noticeable compared to the right preconditioned case, and also because of the nice clustering properties shown in able and Figure 2. he results for FGMRES and inexact application of P are shown in able 8. We observe a reduction in CPU times of nearly an order of magnitude compared to the exact variant, with the best results obtained for γ =. hese reuslts compare favorably with those recently reported in [3]. We conclude this section with a discussion of the robustness of the proposed preconditioners with respect to the parameter α which, as explained in [6], is the main control parameter. We keep the value of β unchanged (β =.5). he values of α used in these tests are given by α(k) =.5 k α c for k =,...,8 and α c = hese are physically meaningful values of α and correspnd to values used for the tests in [6]. he results are reported for the largest problem size only (n = 327,675) in Figure 4; the trend is similar for smaller problem sizes. From these results we can see that the performance of the preconditioners has a weak dependence on α, with a slight increase in the number of iterations for larger values of α. he block triangular preconditioner with γ = seems to be especially robust with respect to α.

13 Block preconditioners for a class of saddle point systems 3 Size FGMRES (γ = ) FGMRES (γ = ) FGMRES (γ = 2) Iters CPU Iters CPU Iters CPU (63).76 (43).9 3(42) (6).39 9(35).23 3(37) (54).599 8(29).35 3(35) (5).45 8(23).668 2(3) (44) (9).253 (26) (4).425 7(8).256 (2) (36) (5).4852 (2) (39) (4).47 (2).5382 able 5 Numerical results for the inexact application of the preconditioner P, different γ. Size γ = γ = γ = 2 Iter CPU Iter CPU Iter CPU able 6 Numerical results for PGMRES with the preconditioner P for different γ (right preconditioning). Size PGMRES Iter CPU able 7 Numerical results for the preconditioner P with γ = (left preconditioning). Size FGMRES (γ = ) FGMRES (γ = ) FGMRES (γ = 2) Iters CPU Iters CPU Iters CPU (3).22 6(23).95 (25) (26).248 5(9).52 (2) (25).427 5(9).295 (9) (24).795 5(9).539 (8) (2).66 5(7).9 (7) (2) (2).23 9(4) (7).66 4().48 9(4) (7).57 4().939 9(4).3257 able 8 Numerical results for the inexact implementation of the preconditioner P, different γ.

14 4 Fatemeh Panjeh Ali Beik, Michele Benzi Block triangular preconditioner Block diagonal preconditioner Fig. 4 Number of iterations for FGMRES (op) and inner PCG (Bottom) versus k; Left to right: γ =, γ = and γ = 2. 6 Conclusions In this paper we have studied two block preconditioners for a class of linear systems arising from the modeling of liquid crystals directors. Bounds for the eigenvalues of the preconditioned matrices have been established, and numerical experiments with both exact and inexact variants of the preconditioners have been performed. he inexact variants, used together with FGMRES, result in rapid convergence independent of mesh size, leading to virtually optimal solvers for the application of interest. Acknowledgements We would like to thank Alison Ramage for providing the test problems used in the numerical experiments. he first author is grateful for the hospitality of the epartment of Mathematics and Computer Science at Emory University, where this work was completed. References. J. H. Adler,. J. Atherton,. R. Benson,. B. Emerson, and S. P. MacLachlan, Energy minimization for liquid crystal equilibrium with electric and flexoelectric effects, SIAM J. Sci. Comput., 37, S57 S76 (25) 2. J. H. Adler,. J. Atherton,. B. Emerson, and S. P. MacLachlan, An energy-minimization finite-element approach to the Frank Oseen model of nematic liquid crystals, SIAM J. Numer. Anal., 53, (25) 3. F. P. A. Beik and M. Benzi, Iterative methods for double saddle point systems, SIAM J. Matrix Anal. Applic., 39, (28) 4. M. Benzi, G. H. Golub, and J. Liesen, Numerical solution of saddle point problems, Acta Numer., 4, 37 (25) 5. R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK (985) 6. A. Ramage and E. C. Gartland, Jr., A preconditioned nullspace method for liquid crystal director modeling, SIAM J. Sci. Comput., 35, B226 B247 (23) 7. Y. Saad, Iterative Methods for Sparse Linear Systems. Second Edition, Society for Industrial and Applied Mathematics, Philadelphia (23) 8. V. Simoncini and. B. Szyld, Recent computational developments in Krylov subspace methods for linear systems, Numer. Linear Algebra Appl., 4, 59 (27)