New variants of deflation techniques for bubbly flow problems

For various applications, it is well-known that deflated ICCG is an efficient method for solving linear systems with invertible and singular co-efficient matrix. This deflated ICCG with subdomain deflation vectors is used by us to solve linear systems with singular coefficient matrix, arising from a discretization of the Poisson equation with Neumann boundary conditions and discontinuous coefficients. However, we have not explained the fact that the use of subdomain deflation vectors is effective. In this paper, we investigate this by doing some perturbed eigenvector analysis and we make it plausible that the use of subdomain deflation vectors is beneficial for the number of iterations and the amount of computational time. Moreover, we introduce new variants of the deflation technique which can deal with bubbly flow problems and with two-phase flow problems in general. The most simple variant is the so-called levelset deflation technique which chooses deflation vectors corresponding to the bubbles in the domain. Another variant is to combine the subdomain and levelset deflation technique which makes profit of the advantages of the original subdomain deflation and the levelset deflation. Numerical experiments show the good performance of these new deflation variants. Finally, the undeflated and deflated MICCG and RICCG methods are also investigated, which apply the modified and relaxed IC preconditioners instead of the original IC preconditioner, respectively. We show that the extension of the deflation technique to these methods is only beneficial for test problems with smooth coefficients. In our bubbly flow problems, the original ICCG and DICCG appear to be the most efficient methods