Unconditionally Stable Finite Difference, Nonlinear Multigrid Simulation of the Cahn-Hilliard-Hele-Shaw System of Equations

We present an unconditionally energy stable and solvable finite difference scheme for the Cahn-Hilliard-Hele-Shaw (CHHS) equations, which arise in models for spinodal decomposition of a binary fluid in a Hele-Shaw cell, tumor growth and cell sorting, and two phase flows in porous media. We show that the CHHS system is a specialized conserved gradient-flow with respect to the usual Cahn-Hilliard (CH) energy, and thus techniques for bistable gradient equations are applicable. In particular, the scheme is based on a convex splitting of the discrete CH energy and is semi-implicit. The equations at the implicit time level are nonlinear, but we prove that they represent the gradient of a strictly convex functional and are therefore uniquely solvable, regardless of time step-size. Owing to energy stability, we show that the scheme is stable in the L-s(infinity) (0, T; H-h(1)) norm, and, assuming two spatial dimensions, we show in an appendix that the scheme is also stable in the L-s(2) (0, T; H-h(2)) norm. We demonstrate an efficient, practical nonlinear multigrid method for solving the equations. In particular, we provide evidence that the solver has nearly optimal complexity. We also include a convergence test that suggests that the global error is of first order in time and of second order in space.