Piecewise-linear, discontinuous Galerkin method for a fractional diffusion equation

We use 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 error in L_∞((0,T);L₂(ω)) is of order k^{2 + α}, where k denotes the maximum time step. Since derivatives of the solution may be singular at t = 0, our result requires the use of non-uniform time steps. In the limiting case α = 0 we recover the known O(k²) convergence for the classical diffusion (heat) equation. We also consider a fully-discrete scheme that employs standard (continuous) piecewise-linear finite elements in space, and show that the additional error is of order h²log(1/k). Numerical experiments indicate that our O(k^{2 + α}) error bound is pessimistic. In practice, we observe O(k²) convergence even for α close to - 1.