Geodesic
Efficient deflation methods applied to 3-D bubbly flow problems
Electronic transactions on numerical analysis, Tome 26 (2007), pp. 330-349.

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Summary: For various applications, it is well-known that deflated ICCG is an efficient method to solve linear systems with an invertible coefficient matrix. Tang and Vuik [J. Comput. Appl. Math., 206 (2007), pp. 603- 614] proposed two equivalent variants of this deflated method, which can also solve linear systems with singular coefficient matrices that arise from the discretization of the Poisson equation with Neumann boundary conditions and discontinuous coefficients. In this paper, we also consider the original variant of DICCG in Vuik, Segal, and Meijerink [J. Comput. Phys., 152 (1999), pp. 385-403], that already proved its efficiency for invertible coefficient matrices. This variant appears to be theoretically equivalent to the first two variants, so that they all have the same convergence properties. Moreover, we show that the associated coarse linear systems within these variants can be solved both directly and iteratively. In applications with large grid sizes, the method with the iterative coarse solver can be substantially more efficient than the one with the standard direct coarse solver.
Classification : 65F10, 65F50, 65N22
Keywords: deflation, conjugate gradient method, preconditioning, Poisson equation, symmetric positive semidefinite matrices, bubbly flow problems, inner-outer iterations 
@article{ETNA_2007__26__a6,
     author = {Tang, J.M. and Vuik, C.},
     title = {Efficient deflation methods applied to {3-D} bubbly flow problems},
     journal = {Electronic transactions on numerical analysis},
     pages = {330--349},
     publisher = {mathdoc},
     volume = {26},
     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2007__26__a6/}
}
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Tang, J.M.; Vuik, C. Efficient deflation methods applied to 3-D bubbly flow problems. Electronic transactions on numerical analysis, Tome 26 (2007), pp. 330-349. http://geodesic.mathdoc.fr/item/ETNA_2007__26__a6/