## Generalized block triangular preconditioner for symmetric saddle point problems.(English)Zbl 1184.65035

V. Simoncini [Appl. Numer. Math. 49, No. 1, 63–80 (2004; Zbl 1053.65033)] and on the other hand Z.-H. Cao [Appl. Numer. Math. 57, No. 8, 899–910 (2007; Zbl 1118.65021)] have discussed two block triangular preconditioners for symmetric saddle point problems. By multiplying the given large dimensional matrix with such a preconditioner the performance of iterative methods such as the generalized minimal residual method (GMRES($$n$$)) is improved dramatically. Here by introdducing an additional parameter, the two above mentioned preconditioners are embedded in a family of preconditioners and will so give room for improvements. For the discussion of the expected numerical behaviour regions where the eigenvalues are situated are described. Numerical experiments for such a problem are described in detail.

### MSC:

 65F08 Preconditioners for iterative methods 65F10 Iterative numerical methods for linear systems

### Citations:

Zbl 1053.65033; Zbl 1118.65021

IFISS
Full Text:

### References:

 [1] Axelsson O, Barker VA (1984) Finite element solution of boundary value problems. Academic Press, Orlando · Zbl 0537.65072 [2] Brezzi F, Fortin M (1991) Mixed and hybrid finite element methods. Springer, New York · Zbl 0788.73002 [3] Elman HC, Silvester DJ, Wathen AJ (2003) Finite elements and fast iterative solvers. Oxford University Press, Oxford [4] Zulehner W (2001) Analysis of iterative methods for saddle point problemsa unified approach. Math Comp 71: 479–505 · Zbl 0996.65038 [5] Benzi M, Golub GH, Liesen J (2005) Numerical solution of saddle point problems. Acta Numer 14: 1–137 · Zbl 1115.65034 [6] Klawonn A (1998) Block-triangluar preconditioners for saddle-point problems. SIAM J Sci Comput 19: 172–184 · Zbl 0917.73069 [7] Silvester DJ, Wathen AJ (1994) Fast iterative solution of stabilized Stokes systems, Part II: using general block preconditioners. SIAM J Numer Anal 31: 1352–1367 · Zbl 0810.76044 [8] Wathen AJ, Fischer B, Silvester DJ (1995) The convergence rate of the minimal residual method for the Stokes problem. Numer Math 71: 121–134 · Zbl 0837.65026 [9] Durazzi C, Ruggiero V (2003) Indefinitely preconditioned conjugate gradient method for large sparse equality and inequatlity constrained quadratic problems. Numer Linear Algebra Appl 10(8): 673–688 · Zbl 1071.65512 [10] Fischer B, Ramage AR, Silvester DJ, Wathen AJ (1998) Minimun reidual methods for augmented systems. BIT 38: 527–543 · Zbl 0914.65026 [11] Ipsen ICF (2001) A note on preconditioning nonsysmmetric matrices. SIAM J Sci Comput 23: 1050–1051 · Zbl 0998.65049 [12] Perugia I, Simoncini V (2000) Block-diagonal and indefinite sysmmetric preconditioners for mixed finite element formulations. Numer Linear Algebra Appl 7: 585–616 · Zbl 1051.65038 [13] Rusten T, Winther R (1992) A preconditioned iterative method for saddle point problems. SIAM J Matrix Anal Appl 13: 887–904 · Zbl 0760.65033 [14] Simoncini V (2004) Block triangular preconditioners for symmetric saddle-point problems. Appl Numer Math 49: 63–80 · Zbl 1053.65033 [15] Johson C, Thomee V (1981) Error estimates for some mixed finite element methods for parabolic type problems. RAIRO Anal Numer 15: 41–78 · Zbl 0476.65074 [16] Murphy MF, Golub GH, Wathen AJ (2000) A note on preconditoning for indefinite linear systems. SIAM J Sci Comput 21: 1969–1972 · Zbl 0959.65063 [17] Saad Y (1996) Iterative methods for sparse linear systems. PWS Publishing Company, Boston · Zbl 1031.65047 [18] Cao ZH (2007) Positive stable block triangular preconditioners for symmetric saddle point problems. Appl Numer Math 57: 899–910 · Zbl 1118.65021 [19] Silvester DJ, Elman HC, Ramage A, IFISS: incompressible flow iterative solution software. http://www.manchester.ac.uk/ifiss
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.