×

FGMRES preconditioning by symmetric/skew-symmetric decomposition of generalized Stokes problems. (English) Zbl 1161.76033

Summary: The generalized Stokes problem is solved for non-standard boundary conditions. This problem arises after time semi-discretization by ALE method of the Navier-Stokes system, which describes the flow of two immiscible fluids with similar densities but different viscosities in a horizontal pipe, when modeling heavy crude oil transportation. We discretized the generalized Stokes problem in space using the “mini”-finite element. The inf-sup condition is proved when the interface between the two fluids and its discretization match exactly. The linear system obtained after discretization is solved using different iterative Krylov methods with and without preconditioning. Numerical experiments with different meshes are presented as well as comparisons between the methods considered. The results suggest that FGMRES and a preconditioning technique based on symmetric/skew-symmetric decomposition is a promising candidate for solving large scale generalized Stokes problem.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
65F10 Iterative numerical methods for linear systems

Software:

UCSparseLib
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Arrow, K.J.; Hurwicz, L.; Arrow, K.J.; Hurwicz, L.; Uzawa, H., (), 117-126
[2] Ashcraft, C.; Grimes, R.; Lewis, J.G., Accurate symmetric indefinite linear equation solvers, SIAM J. matrix anal. appl., 20, 513-561, (1998) · Zbl 0923.65010
[3] Atkinson, K.E.; Han, W., Theoretical numerical analysis: A functional analysis framework, vol. 39 of text in applied mathematics, (2001), Springer-Verlag New York
[4] Bai, Z.Z.; Golub, G.H.; Ng, M.K., Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. matrix anal. appl., 24, 603-626, (2003) · Zbl 1036.65032
[5] Bank, R.E.; Welfert, B.D.; Yserentant, H., A class of iterative methods for solving saddle point problems, Numer. math., 56, 645-666, (1990), MR91b:65035 · Zbl 0684.65031
[6] Barnard, S.T.; Grote, M.J., A block version of SPAI preconditioner, ()
[7] Benzi, M., Preconditioning techniques for large linear system: A survey, J. comp. phys., 182, 4, 418-477, (2002) · Zbl 1015.65018
[8] Benzi, M.; Golub, G.H., A preconditioner for generalized saddle point problems, SIAM J. matrix anal. appl., 26, 1, 20-41, (2004) · Zbl 1082.65034
[9] Benzi, M.; Golub, G.H.; Liesen, J., Numerical solution of saddle point problems, Acta numer., 14, 1-137, (2005) · Zbl 1115.65034
[10] Bertsekas, D.P., Nonlinear programming, (1999), Athena Scientific Nashua NH · Zbl 0935.90037
[11] Boland, I.; Nicolaides, R., Stability of finite elements under divergence constraints, SIAM J. numer. anal., 20, 722-731, (1983) · Zbl 0521.76027
[12] Bramble, J.H.; Pasciak, J.E., A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems, Math. comp., 50, 1-17, (1988) · Zbl 0643.65017
[13] Brezzi, F.; Fortin, M., Mixed and hybrid finite element method, vol. 15 of Springer series in computational mathematics, (1991), Springer-Verlag New York
[14] Cao, Z., Fast Uzawa algorithm for generalized saddle point problems, Appl. numer. math., 46, 2, 157-171, (2003) · Zbl 1032.65029
[15] Chow, E.; Saad, Y., Approximate inverse techniques for block-partitioned matrices, SIAM J. sci. comput., 18, 6, 1657-1675, (1997) · Zbl 0888.65035
[16] Ciarlet, P.G., The finite element method for elliptic problems, vol. 4, (1978), North-Holland Company · Zbl 0445.73043
[17] De Cecchis, D., Método precondicionado para la simulación de flujo bifásico en tuberías horizontales, trabajo de grado de maestría, (2006), Universidad Central de Venezuela Caracas, Venezuela
[18] De Cecchis, D.; López, H.; Molina, B.; Gámez, B.; Ojeda, D.; Larrazábal, G.; Cerrolaza, M., Preconditioning techniques of the Krylov methods for the discretization of Stokes generalized problem (in Spanish)/simulación y modelado en ingeniería y ciencias, sociedad venezolana de Métodos numéricos en ingeniería, (2006), SVMNI
[19] Eisenstat, S.C.; Elman, H.C.; Schultz, M.H., Variational iterative methods for nonsymmetric systems of linear equations, SIAM J. numer. anal., 20, 345-357, (1983) · Zbl 0524.65019
[20] Elman, H.C., Preconditioners for saddle point problems arising in computational fluid dynamics, Appl. numer. math., 43, 75-89, (2002) · Zbl 1168.76348
[21] Fortin, M., Finite element solution of the navier – stokes equations, Acta numer., 239-284, (1993) · Zbl 0801.76043
[22] Fortin, M.; Fortin, A., A generalization of uzawa’s algorithm for the solution of the navier – stokes equations, Comm. appl. numer. meth., 1, 205-208, (1985) · Zbl 0592.76040
[23] Freund, R.; Nachtigal, N.M., QMR: A quasi-minimal residual method for non-Hermitian linear systems, Numer. math., 60, 315-339, (1991) · Zbl 0754.65034
[24] Girault, V.; López, H.; Maury, B., One time-step finite element discretization of the equation of motion of two-fluid flows, Numer. meth. part. diff. eq., 22, 3, 680-707, (2006) · Zbl 1089.76032
[25] Girault, V.; Raviart, P., Finite element methods for navier – stokes equations. theory and algorithms, (1986), Springer-Verlag Berlin, Germany · Zbl 0585.65077
[26] Greenbaum, A., Iterative methods for solving linear system, vol. 17 of frontiers in applied mathematics, (1997), SIAM
[27] Haws, J.C.; Meyer, C.D., Preconditioning KKT systems, Numer. linear algebra appl., 00, 1-6, (2001)
[28] Hestenes, M.R.; Stiefel, E.L., Methods of conjugate gradients for solving linear systems, J. res. natl. bur. stand., 49, 409-436, (1952) · Zbl 0048.09901
[29] Hiptmair, R.; Hoppe, R.H.W., Multilevel methods for mixed finite elements in three dimensions, Numer. math., 82, 253-279, (1999) · Zbl 0929.65124
[30] Krzyżanowski, P., On block preconditioners for nonsymmetric saddle point problems, SIAM J. sci. comput., 23, 1, 157-169, (2001) · Zbl 0998.65048
[31] Larrazábal, G., Ucsparselib: A numerical library to solve sparse linear systems, in: numerical simulation computational modeling, Svmni, TC19-TC25, (2004)
[32] Li, J.; Renardy, Y., Numerical study of flows of two immiscible liquids at low Reynolds number, SIAM rev., 42, 3, 417-439, (2000) · Zbl 0981.76062
[33] Maury, B., Characteristics ALE method for the unsteady 3D navier – stokes equations whith a free surface, Int. J. comp. fluid dyn., 6, 175-188, (1996)
[34] Paige, C.C.; Saunders, M.A., Solution of sparse indefinite system of linear equations, SIAM J. numer. anal., 12, 617-629, (1975) · Zbl 0319.65025
[35] Rusten, T.; Winther, R., A preconditioned iterative method for saddle point problems, SIAM J. matrix anal. appl., 13, 887-904, (1992) · Zbl 0760.65033
[36] Saad, Y., A flexible inner-outer preconditioned gmres algorithm, SIAM J. sci. comput., 14, 2, 461-469, (1993) · Zbl 0780.65022
[37] Saad, Y.; Schultz, M.H., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. sci. stat. comp., 7, 856-869, (1986) · Zbl 0599.65018
[38] Silvester, D.; Elman, H.; Kay, D.; Wathen, A., Efficient preconditioning of linearized navier – stokes equations for incompressible flow, J. comp. appl. math., 128, 261-279, (2001) · Zbl 0983.76051
[39] Silvester, D.; Wathen, A., Fast iterative solution of stabilized Stokes systems. part II: using block preconditioners, SIAM J. numer. anal., 31, 1352-1367, (1994) · Zbl 0810.76044
[40] Simoncini, V., Block triangular preconditioners for symmetric saddle-point problems, Appl. numer. math., 49, 1, 63-80, (2004) · Zbl 1053.65033
[41] Simoncini, V.; Benzi, M., Spectral properties of the Hermitian and skew-Hermitian splitting preconditioner for saddle point problems, SIAM J. matrix anal. appl., 26, 2, 377-389, (2004) · Zbl 1083.65047
[42] Sturler, R.D.; Liesen, J., Block-diagonal and constraint preconditioners for nonsymmetric indefinite linear systems. part i: theory, SIAM J. sci. comput., 26, 5, 1598-1619, (2005) · Zbl 1078.65027
[43] Uzawa, H.; Arrow, K.J.; Hurwicz, L.; Uzawa, H., (), 154-165
[44] Verfürth, R., A combined conjugate gradient-multigrid algorithm for the numerical solution for the Stokes problem, IMA J. numer. anal., 4, 441-455, (1984) · Zbl 0563.76028
[45] Wright, M.H., Interior point methods for constrained optimization, Acta numer., 341-407, (1992) · Zbl 0766.65053
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.