Linearized numerical schemes for nonlocal Cahn-Hilliard equation and its convergence analysis

A stabilized linear semi-implicit numerical scheme is considered for the nonlocal Cahn-Hilliard equation, and a detailed convergence analysis is presented. This theoretical analysis follows from consistency and stability estimates for the numerical error function. Due to the complicated form of the nonlinear term, we adopt the discrete H^{-1} norm for the error function to establish the convergence result. In addition, an assumption on the uniform maximum bound of the numerical solution is required for the theoretical justification of the energy stability, and such a bound is derived by conducting the higher order consistency analysis. Taking the view that the numerical solution is indeed the exact solution with a perturbation, the error function is uniformly bounded under a loose constraint of the time step size, which then leads to the uniform maximum-norm bound of the numerical solution. The second order accurate numerical schemes, either in the BDF2 or Crank-Nicolson approximation, are also analyzed as well.