Finite volume element method for two-dimensional fractional subdiffusion problems

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 + k^1+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.