The time-fractional Cahn-Hilliard equation: analysis and approximation

We consider a time-fractional Cahn-Hilliard equation where the conventional first-order time derivative is replaced by a Caputo fractional derivative of order $\alpha \in (0,1)$. Based on an a priori bound of the exact solution, global existence of solutions is proved and detailed regularity results are included. A finite element method is then analyzed in a spatially discrete case and in a completely discrete case based on a convolution quadrature in time generated by the backward Euler method. Error bounds of optimal order are obtained for solutions with smooth and nonsmooth initial data, thereby extending earlier studies on the classical Cahn-Hilliard equation. Further, by proving a new result concerning the positivity of a discrete time-fractional integral operator, it is shown that the proposed numerical scheme inherits a discrete energy dissipation law at the discrete level. Numerical examples are presented to illustrate the theoretical results.