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 l(2)(0; T; H-h(3)) stability of the numerical scheme. To overcome the difficulty associated with the convection term del . (phi u), we perform an l(infinity)(0; T; H-h(1)) error estimate instead of the classical l(infinity)(0; T; l(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.