ON A GENERAL CLASS OF PRECONDITIONERS FOR NONSYMMETRIC GENERALIZED SADDLE POINT PROBLEMS

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1 U..B. Sci. Bull., Series A, Vol. 78, Iss. 4, 06 ISSN ON A GENERAL CLASS OF RECONDIIONERS FOR NONSYMMERIC GENERALIZED SADDLE OIN ROBLE Fatemeh anjeh Ali BEIK his paper deals with applying a class of preconditioners for saddle point problems with nonsymmetric positive definite (,) part symmetric positive semidefinite (,) part. he examined class of preconditioners includes the generalized shift-splitting preconditioner their modified version. he offered class of preconditioner is induced from a stationary iterative method. Numerical experiments for a model Navier-Stokes problem are reported which illustrate the efficiency of the presented preconditioner. Keywords: Nonsymmetric saddle point problem, preconditioner, shift-splitting, iterative method. C00: 65F0, 65F50 65N.. Introduction Consider generalized saddle point problems of the form A B x f A u b, B C y g () such that A nn is nonsymmetric positive definite (i.e., the symmetric part ( A A ) is positive definite; B mn is of full row rank ( m n) C is symmetric positive semidefinite. In [, Lemma.], it has been shown that the preceding assumptions guarantee existence uniqueness of the solution of (). Linear systems of the form () arise in a variety of computational sciences engineering applications such as constrained optimization, computational fluid dynamics, mixed finite element discretization of the Navier-Stokes equations, the linear elasticity problem, elliptic parabolic interface problems, constrained least-squares problem so on; for more details see [,, 3, 9, 0] references therein. In the past few years, there has been a growing interest in saddle problems of the form () several types of iterative methods have been proposed; we refer the reader to [] for a comprehensive survey. More precisely, recently, some research works have focused on variants types of the Uzawa method for the case that (,)-block is zero; for more details see [4, 5, 6] references therein. Vali-e-Asr University of Rafsanjan, Iran, Islamic Republic Of, beik.fatemeh@gmail.com

2 Fatemeh anjeh Ali Beik More recently, Liang Zhang [] have presented a modified Uzawa method based on the skew-ermitian triangular splitting (SS) of the (,) part of the saddle-point coefficient matrix for solving non-ermitian saddle-point problems with non-ermitian positive definite skew-ermitian dominant (,) zero (,) part. It is well-known that the Krylov subspace methods are superior to the stationery iterative methods in general. Nevertheless, the Krylov subspace methods converge slowly for solving saddle point problems in practice. o overcome this drawback, several papers have applied different types of preconditioners till now. For instance, Cao et al. [4] have proposed a modified dimensional split preconditioner based on a splitting of the generalized saddle point matrix. In the case that the (,)-block C is zero the (,)-block is symmetric positive definite. In [5], Cao et al. have examined the shift-splitting preconditioner of the form I A B SS, 0. () B I which is obtained based on shift-splitting of the saddle point coefficient matrix. Also the application of a local preconditioner is studied A B LSS, 0. B I Recently, Chen Ma [7] have focused on the saddle point problems with the similar form mentioned in [5] developed a general class of preconditioners which incorporates (). More precisely, the following preconditioner has been offered I A B GSS, (3) B I where 0 0. Lately, Salkuyeh et al. [3] have applied a modified generalized shiftsplitting (MGSS) preconditioner I A B MGSS,, 0, (4) B I C for the saddle point problems of the form () so that the (,)-block is symmetric positive definite C 0. he MGSS preconditioner is based on a splitting of the saddle point coefficient matrix which results in an unconditionally convergent stationary iterative method. In addition, the following relaxed version of the MGSS preconditioner is examined

3 On a general class of preconditioners for nonsymmetric generalized saddle point problems 3 RMGSS A B, 0. B I C More recently, Cao et al. [6] have studied the application of the generalized shift-splitting preconditioner (3) for saddle point problem () whose (,)-block is zero (,)-block is a nonsymmetric positive definite matrix. In this paper, we introduce a class of general preconditioners which are obtained by splitting the diagonal blocks of the coefficient matrix A in (). o do this, first, the mixed splitting () iterative method is presented its convergence properties are discussed. he offered iterative method includes the modified generalized shift-splitting (MGSS) iterative method [3]. Notice that the convergence properties of the MGSS iterative method the application of the preconditioners of the form (4) have not been studied for the case that the (,)-block A in () is nonsymmetric. As seen, the induced preconditioner by the method incorporates the preconditioners (), (3) (4). he rest of this paper is organized as follows. Before ending this section, some notations are introduced which are exploited throughout the paper. In Section, first, we present the mixed splitting () iterative method give some sufficient conditions under which the method is convergent. hen based on the presented splitting, a general class of preconditioners is offered to improve the speed of convergence of the Krylov subspace methods to solve the saddle point problems of the form (). In Section 3, some numerical experiments are reported for a model of Navier-Stokes problem to illustrate the efficiency of the proposed new class of preconditioners. Finally the paper is finished with a brief conclusion in Section 4. Notations: he real conjugate parts of a complex number a are denoted by Re ( a) a, respectively. For a square matrix A, the symbol ( A) sts for the spectral radius of A. For two given symmetric positive definite matrices A B, the notation A B means that A B is positive definite.. Main results Consider the following mixed splitting for the coefficient matrix A, ˆ ˆ M B N B A Q, (5) B M B N in which ˆM M are both nonsingular matrices such that A Mˆ Nˆ C M N. Remark.. Assume that C 0. Consider a special case of (5) such that

4 4 Fatemeh anjeh Ali Beik ˆ ˆ M ( I A), N ( I A), M I N I, where ( )0 0. Evidently, in this case, the mixed splitting reduces to the generalized shift-splitting considered in [6]. Note that it is not difficult to verify that Mˆ Nˆ is symmetric (semi) positive definite, M N is nonsingular, M is symmetric positive definite ˆ ( M Nˆ) ( ). Remark.. Suppose that ( ) 0 0 are given A is symmetric. We comment here that setting ˆ ˆ M ( I A), N ( I A), M I C N I C, in (5), we obtain the modified generalized shift-splitting splitting discussed in [3]. Similar to the previous remark, we point out that here Mˆ Nˆ is symmetric (semi) positive definite, M N is nonsingular, M is symmetric positive definite ˆ ( M Nˆ) ( ). Mixed splitting () iterative method: Exploiting the mixed splitting (5), we may define the iterative scheme to solve () as follows: ( k) ( k) u u c, k 0,,,, (6) where Q is the iteration matrix, c b, u (( x ),( y ) ) ( k ) ( k ) ( k ) (0) the initial guess u is given. It is well-known that (6) is convergent for any (0) arbitrary initial guess u iff ; for further details one may refer to [, Chapter 4]. In the sequel we discuss the convergence of the method. Let be an arbitrary eigenvalue of u ( x, y ) be its corresponding eigenvector. u u Q which is equivalent to say that u hat is, Q u. ence, we obtain ˆ ˆ Nx B y Mx B y, (7) Bx Ny Bx My. (8) Lemma.. Suppose that M N is nonsingular. If is an eigenvalue of, then.

5 On a general class of preconditioners for nonsymmetric generalized saddle point problems 5 roof. Assume that u is the corresponding eigenvector of, hence u 0. In view of the fact that A is nonsingular then. Since implies that A u 0. From (8), we find that if then ( M N) u 0 which is a contradiction by our assumption that M N is nonsingular. Remark.3. In view of Remarks.., throughout this work, we assume that Mˆ Nˆ is symmetric positive semidefinite, M N is nonsingular, M is symmetric positive definite, M C ˆ ( M Nˆ). heorem.. Assume that denotes the iteration matrix of the method. Suppose that Mˆ Nˆ is symmetric positive semidefinite, M N is nonsingular, M is symmetric positive definite, M C ˆ ( M Nˆ). hen. roof. Let (, u) be an arbitrary eigenpair of so that 0 where u ( x, y ). We first show that x 0. If x 0 then from (7), it is seen that ( ) B y 0. Now Lemma. implies that y 0, hence u 0 which cannot be true invoking the fact that u is an eigenvector. Notice if y 0 then from (7), we have Nx ˆ Mx ˆ. hat is is an eigenvalue of Mˆ Nˆ. herefore, by the assumption. In the rest of the proof without loss of generality, it is assumed that x 0 y 0. From Eqs (7) (8), we have ˆ x B y x Mx x ( Mˆ Nˆ ) x, (9) y B x y My y Cy. (0) Note that the left h side of (9) is equal to the conjugate of the left h side of (0), hence ˆ ˆ ˆ x M x x ( M N ) x y My y Cy. Or equivalently, ˆ ˆ ˆ x M x x ( M N) x y My y Cy. For simplicity, we set ( ˆ ˆ q x M N) x, r y My s y Cy. Now from the above relation, it is seen that

6 6 Fatemeh anjeh Ali Beik Re ˆ x M x q s Re. () r s q We point out that by the assumptions, rs, q are positive r s. On the other h, straightforward computations reveal that ˆ Re Re ˆ x M x q x Mx x ( Mˆ Nˆ ) x Re( x Ax) 0. Consequently, from (), we deduce that Re 0. Now the result follows immediately from the following fact Re. Remark.4. As pointed earlier, Salkuyeh et al. [3] has developed the MGSS method for saddle point problems with symmetric positive definite (,)-block. It is pointed in Remark. that the MGSS method is a special case of the method. herefore, heorem. turns out that all of the results established by Salkuyeh et al. are valid when (,)-block A is nonsymmetric positive definite. Remark.5. Assume that Ais a nonsymmetric positive definite matrix. Let Q be two arbitrary symmetric positive definite matrices. If we set Mˆ A, Nˆ A, M ( Q C) N ( Q C), then it can be seen that all of the assumptions in heorem. hold. Our numerical experiments illustrate that the matrix can be served as an effective preconditioner to speed up the convergence of the Krylov subspace methods (e.g., restarted GMRES method) although its corresponding stationery iterative method may converge slowly. he same ideas are utilized in [5, 6, 7, 3] which focus on the shift-splitting types of preconditioners. o apply the preconditioner induced from the mixed splitting, in fact, we need to hle the Krylov subspace methods for solving the following linear system Au b. Remark.6. In view of Remark.5, in our numerical experiments, we set Mˆ A, Nˆ A, M ( I C) N ( I C), where are two given positive parameters is the symmetric part of A. hat is, we use the following preconditioner

7 On a general class of preconditioners for nonsymmetric generalized saddle point problems 7 * A B. () B I C he following proposition presents the eigenvalue distribution of the * ( ) A. * roposition.. Assume that is defined as in (). If A is nonsymmetric * positive definite, then the eigenvalues of ( ) A belongs to the following subset of, S { z iw z 0, z w 4 w z}. * roof. Let i be an arbitrary eigenvalue of ( ) A. herefore, * Au u, (3) so that u ( x, y ). It is not difficult to verify that 0 x 0. From (3), it can be observed that x Ax x B y x Ax x x x B y, y Bx y Cy y Bx y Cy y y, where /. Consequently, we have x B y x Ax x x, y Bx y Cy y y. For simplicity, assume that, x Ax p ip q y Cy r y y. he above two relations implies that ( p ip) p q r. Since A is positive definite ( p 0), it can be concluded that Re( ) he preceding relation shows that Re( ) Im( ), we deduce that p r p 0. Re( ). In view of the fact that Re( ) Im( ) which complete the proof.

8 8 Fatemeh anjeh Ali Beik 3. Numerical experiments In this section, we give an example to demonstrate the applicability of the proposed preconditioner (in the form given in Remark.6) compare its performance with the shift-splitting types. All of the reported experiments were performed on a 64-bit.45 Gz core i7 processor 8.00GB RAM using some matlab codes on MALAB version In matrices GSS, DSS, MGSS, RMGSS parameters, are taken to be Example 3.. Consider the Oseen equation as follows: w ( v. ) w p f in. w 0. *, the corresponding which is derived when the steady-state Navier-Stokes equation is linearized by icard iteration. ere the vector field v is the approximation of w from the previous icard iteration, the vector w represents the velocity, p denotes the pressure is a bounded domain. he parameter 0 sts for the viscosity. Conservative discretizations of the Oseen problem (4) leads to a generalized saddle point system of the form () with nonsymmetric positive definite (,)-block. Our examined test problems are constructed using IFISS software package written by Elman et al. [8]. he package is used to generate discretizations of leaky lid driven cavity problem using stabilized Q-0 (Q-Q) finite elements method (FEM) to study the performance of the offered preconditioner for the case that (,)-block in () is nonzero (zero). Four grids are exploited, 6 6, 3 3, which Q-Q FEM corresponds (578,8), (78,89), (8450,089) (338,45) to ( nm, ), respectively Q-0 FEM corresponds (578,89), (78,04), (8450, 4096) (338,6384) to ( nm, ), respectively. In all of the tests, we apply the GMRES(5) method such that the initial guess is taken to be zero the iterations are terminated as soon as ( k ) 6 b A u 0 b, or if the number of iterations exceeds from kmax 5000 where ( k ) u (4) denotes the k th approximate solution. he corresponding results are disclosed in ables. We end this section with a remark given as follows. Remark 3.. In [3], the application of the MGSS the relaxed MGSS (RMGSS) preconditioners only studied under the assumption that the (,)-block A is symmetric positive definite. Our numerical tests show that this

9 On a general class of preconditioners for nonsymmetric generalized saddle point problems 9 preconditioners are also effective when Ais nonsymmetrical positive definite. ere, we point out that the proposed MGSS iterative scheme is not convergence in the case that the parameter 0, i.e., when MGSS reduces to RMGSS. On the h, the RMGSS preconditioner outperforms the MGSS preconditioner which the reason can be expressed as in [3, Remark ]. As seen, is as effective as RMGSS this may be a motivation to study the other kind of preconditioners in the form of in the future works with details. able Numerical results for Oseen problem with ν = 0.0 (Q-Q FEM) reconditioner I MGSS RMGSS able Numerical results for Oseen problem with ν = 0.0 (Q-0 FEM) * * Grid I CU I CU I CU I CU reconditioner I GSS DSS * Grid I CU I CU I CU I CU Conclusions Recently, Cao et al. [Appl. Math. Lett. 49 (05), 0-7] Salkuyeh et al. [Appl. Math. Lett. 48 (05), 55-6] have focused on applying shift-splitting preconditioners for saddle point problems. his paper has been concerned with employing a general class of preconditioners incorporates shift-splitting preconditioners. Our established results have revealed that the results proved by Salkuyeh et al. are valid for the case that (,) part of the coefficient matrix of the saddle point problem is nonsymmetric. he reported numerical results by Cao et al. (Salkuyeh et al.) shown that the proposed DSS (RMGSS) preconditioner outperforms their other hled preconditioners although the corresponding

10 0 Fatemeh anjeh Ali Beik iterative method to DSS (RMGSS) preconditioner is not convergent. Our examined numerical experiments have demonstrated that a special preconditioner of the offered class is as effective as DSS RMGSS preconditioners. Further work can be focused on studying the performance of other possible instances of the introduced class of the preconditioners with detail. R E F E R E N C E S []. M. Benzi G.. Golub, A preconditioner for generalized saddle point problems, in SIAM J. Matrix Anal. Appl., vol. 6, no., 004, pp []. M. Benzi, G.. Golub J. Liesen, Numerical solution of saddle point problems, in Acta Numer. vol. 4, no., 005, pp [3]. F. Brezzi M. Fortin, Mixed ybrid Finite Element Methods, Springer-Verlag, New York, 99. [4]. Y. Cao, L. Q. Yao M. Q. Jiang, A modified dimensional split preconditioner for generalized saddle points problems, J. Appl. Math. Comput., vol. 50, Oct. 03, pp [5]. Y. Cao, J. Du Q. Niu, Shift-splitting preconditioners for saddle point problems, in J. Appl. Math. Comput., vol. 7, Dec. 05, pp [6]. Y. Cao, S. Li L. Q. Yao, A class of generalized shift-splitting preconditioners for nonsymmetric saddle point problems, in Appl. Math. Lett., vol. 49, Nov. 05, pp [7]. C. Chen C. Ma, A generalized shift-splitting preconditioners for nonsymmetric saddle point problems, in Appl. Math. Lett., vol. 43, May 05, pp [8]..C. Elman, A. Ramage, D.J. Silvester, IFISS: A Matlab toolbox for modelling incompressibe flow, in ACM rans. Math. Software., vol. 33, Jun. 007, Article 4. [9]. V. Girault. A. Raviart, Finite Element Methods for Navier Stokes Equations, Springer Verlag, Berlin, 986. [0]. R. Glowinski. Le allec, Augmented Lagrangian Operator-splitting Methods in Nonlinear Mechanics. SIAM, hiladelphia, 989. []. Z. Z. Liang G. F. Zhang, U-SS method for non-ermitian saddle-point problems, in Appl. Math. Lett., vol. 46, Aug. 05, pp. -6. []. Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, hiladelphia, 003. [3]. D. K. Salkuyeh, M. Masoudi D. ezari, On the generalized shift-splitting preconditioner for saddle point problems, in Appl. Math. Lett., vol. 48, Oct. 05, pp [4]. A. L. Yang Y. J. Wu, he Uzawa SS method for saddle-point problems, in Appl. Math. Lett., vol. 38, Dec. 04, pp [5]. J.. Yun, Variants of the Uzawa method for saddle point problem, in Comput. Math. Appl., vol. 65, Apr. 03, pp [6]. G. F. Zhang, J. L. Yang, S. S. Wang, On generalized parameterized inexact Uzawa method for a block two-by-two linear system, in J. Comput. Appl. Math., vol. 55, Jan. 04, pp