A Second Order Energy Stable Scheme for the Cahn-Hilliard-Hele-Shaw   Equations
  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 an $\ell^2 (0,T; H_h^3)$ stability of the numerical scheme. To overcome the difficulty associated with the convection term $\nabla \cdot (\phi \boldsymbol{u})$, we perform an $\ell^\infty (0,T; H_h^1)$ error estimate instead of the classical $\ell^\infty (0,T; \ell^2)$ 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. 
Wenbin Chen and Wenqiang Feng and Yuan Liu and Cheng Wang and Steven   M. Wise
math.NA