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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. [ABSTRACT FROM AUTHOR]

In this article, a semidiscrete spatial finite volume (FV)method is proposed and analyzed for approximating solutions of anomalous subdiffusion equations involving a temporal fractional derivative of order α ∈ (0, 1) in a two-dimensional convex domain. An optimal error estimate in the L∞ (L²)-norm is shown to hold. A superconvergence result is proved, and as a consequence, it is proved that a quasi-optimal order of convergence in the L∞ (L∞)-norm holds. We also consider a fully discrete scheme that employs a FV method in space and a piecewise linear discontinuous Galerkin method to discretize in time. It is further shown that the convergence rate is of order h² + k1+α, where h denotes the spatial discretization parameter and k represents the temporal discretization parameter. Numerical experiments indicate optimal convergence rates in both time and space, and also illustrate that the imposed regularity assumptions are pessimistic. [ABSTRACT FROM AUTHOR]