Deflated Iccg Method Solving the Poisson Equation Derived from 3-d Multi-phase Flows

Simulating bubbly flows is a very popular topic in CFD. These bubbly flows are governed by the Navier-Stokes equations. In many popular operator splitting formulations for these equations, solving the linear system coming from the discontinuous Poisson equation takes the most computational time, despite of its elliptic origins. Sometimes these singular linear systems are forced to be invertible leading to a worse (effective) condition number. If ICCG is used to solve this problem, the convergence is significantly slower than for the case of the original singular problem. In this paper, we show that applying the deflation technique, which leads to the DICCG method, remedies the worse condition number and the worse convergence of ICCG. Moreover, some useful equalities are derived from the deflated variants of the singular and invertible matrices, which are also generalized to preconditioned methods. It appears that solving the invertible and singular linear systems with DICCG leads to exactly the same convergence results. Numerical experiments considering air-bubbles in water emphasize these theoretical results. This means that the deflation method is well-applicable for singular linear systems. In addition, from the numerical experiments it appears that DICCG is insensitive for the geometry of the density field, which is an important advantage of the deflation method.