Efficient preconditioners for the co-incompressible two-phase mixture saddle-point system

Develop efficient projection-based or block factorization preconditioners for the saddle-point linear system arising from the discretization of the co-incompressible two-phase mixture equations, improving upon the computational expense of the geometric multigrid preconditioner.

Background

The linear system formed by the discretized momentum and co-incompressibility constraints is a saddle-point problem. The authors successfully use a geometric multigrid preconditioner with box relaxation on adaptive meshes, but report that it is computationally expensive.

They explicitly state that efficient projection or block factorization preconditioners for this specific two-phase system have not yet been developed, identifying a clear algorithmic gap.