Abstract
We present a second-order-in-time finite difference scheme for the Cahn-Hilliard-Hele-Shaw equations. This numerical method is uniquely solvable and unconditionally energy stable. At each time step, this scheme leads to a system of nonlinear equations that can be efficiently solved by a nonlinear multigrid solver. Owing to the energy stability, we derive anℓ² (0,T; H_(h)³)stability of the numerical scheme. To overcome the difficulty associated with the convection term∇ ⋅ (φ \boldsymbol{u}{)}{} , we perform anℓ^(∞) (0,T; H_(h)¹)error estimate instead of the classicalℓ^(∞) (0,T; ℓ²)one to obtain the optimal rate convergence analysis. In addition, various numerical simulations are carried out, which demonstrate the accuracy and efficiency of the proposed numerical scheme.