Uniform convergence for a discontinuous Galerkin, time-stepping method applied to a fractional diffusion equation

We consider a piecewise linear, discontinuous Galerkin method for the time discretization of a fractional diffusion equation involving a parameter in the range -1 < α < 0. Our analysis shows that, for a time interval (0, T) and a spatial domain Ω, the uniform error in L_∞((0, T); L₂(Ω)) is of order k^ρ, where ρ = min(2,5/2 + α) and k denotes the maximum time step. Thus, if -1/2 ≤ α < 0, then we have optimal O(k²) convergence, just as for the classical diffusion (heat) equation.