Time-stepping discontinuous Galerkin methods for fractional diffusion problems

Time-stepping -versions discontinuous Galerkin (DG) methods for the numerical solution of fractional subdiffusion problems of order with will be proposed and analyzed. Generic -version error estimates are derived after proving the stability of the approximate solution. For -version DG approximations on appropriate graded meshes near , we prove that the error is of order , where is the maximum time-step size and is the uniform degree of the DG solution. For -version DG approximations, by employing geometrically refined time-steps and linearly increasing approximation orders, exponential rates of convergence in the number of temporal degrees of freedom are shown. Finally, some numerical tests are given.