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 < alpha < 0. Our analysis shows that, for a time interval (0, T) and a spatial domain Omega , the uniform error in L sub( infinity )((0, T); L sub(2)( Omega )) is of order k super( rho ), where rho = ming(2, + alpha ) and k denotes the maximum time step. Thus, if -1/2 less than or equal to alpha < 0, then we have optimal O(k super(2)) convergence, just as for the classical diffusion (heat) equation.