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International Journal for Multiscale Computational Engineering
IF: 1.016 5-Year IF: 1.194 SJR: 0.554 SNIP: 0.82 CiteScore™: 2

ISSN Print: 1543-1649
ISSN Online: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.v6.i1.20
pages 13-24

Fast Deflation Methods with Applications to Two-Phase Flows

J. M. Tang
Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft Institute of Applied Mathematics, J. M. Burgerscentrum, Mekelweg 4,2628 CD Delft, The Netherlands
Cornelis Vuik
Delft Institute of Applied Mathematics, Delft University of Technology, Van Mourik Broekmanweg 6, 2628 XE Delft, the Netherlands

ABSTRACT

Traditional Krylov iterative solvers, such as the preconditioned conjugate gradient method, can be accelerated by incorporating a second-level preconditioner. We use deflation as a second-level preconditioner, which is very efficient in many applications. In this paper, we give some theoretical results for the general deflation method applied to singular matrices, which provides us more insight into the properties and the behavior of the method. Moreover, we discuss stability issues of the deflation method and consider some ideas for a more stable method. In the numerical experiments, we apply the deflation method and its stabilized variant to singular linear systems derived from two-phase bubbly flow problems. Because of the appearance of bubbles, those linear systems are ill-conditioned, and therefore, they are usually hard to solve using traditional preconditioned Krylov iterative methods. We show that our deflation methods can be very efficient to solve the linear systems. Finally, we also investigate numerically the stability of these methods by examining the corresponding inner-outer iterations in more detail.


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