Numerical and computational efficiency of solvers for two-phase problems

We consider two-phase flow problems, modelled by the Cahn-Hilliard equation. In this work, the nonlinear fourth-order equation is decomposed into a system of two coupled second-order equations for the concentration and the chemical potential. We analyse solution methods based on an approximate two-by-two block factorization of the Jacobian of the nonlinear discrete problem. We propose a preconditioning technique that reduces the problem of solving the non-symmetric discrete Cahn-Hilliard system to a problem of solving systems with symmetric positive definite matrices where off-the-shelf multilevel and multigrid algorithms are directly applicable. The resulting solution methods exhibit optimal convergence and computational complexity properties and are suitable for parallel implementation. We illustrate the efficiency of the proposed methods by various numerical experiments, including parallel results for large scale three dimensional problems.