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

Abstract

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 $L_\infty\bigr((0,T);L_2(\Omega)\bigr)$ 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(k ²) 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 h ²log(1/k). Numerical experiments indicate that our O(k ^2 + α) error bound is pessimistic. In practice, we observe O(k ²) convergence even for α close to − 1.