An efficient fully-discrete local discontinuous Galerkin method for the Cahn–Hilliard–Hele–Shaw system

In this paper, we develop an efficient and energy stable fully-discrete local discontinuous Galerkin (LDG) method for the Cahn–Hilliard–Hele–Shaw (CHHS) system. The semi-discrete energy stability of the LDG method is proved firstly. Due to the strict time step restriction (Δt=O(Δx4)Δt=O(Δx4)) of explicit time discretization methods for stability, we introduce a semi-implicit time integration scheme which is based on a convex splitting of the discrete Cahn–Hilliard energy. The unconditional energy stability has been proved for this fully-discrete LDG scheme. The fully-discrete equations at the implicit time level are nonlinear. Thus, the nonlinear Full Approximation Scheme (FAS) multigrid method has been applied to solve this system of algebraic equations, which has been shown the nearly optimal complexity numerically. Numerical results are also given to illustrate the accuracy and capability of the LDG method coupled with the multigrid solver.