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We consider the initial/boundary value problem for the fractional diffusionand diffusion-wave equations involving a Caputo fractional derivative in time.We develop two "simple" fully discrete schemes based on the Galerkin finiteelement method in space and convolution quadrature in time with the generatingfunction given by the implicit backward Euler method/second-order backwarddifference method, and establish error estimates optimal with respect to theregularity of the initial data. These two schemes are first and second-orderaccurate in time for nonsmooth initial data. Extensive numerical experimentsfor one and two-dimensional problems confirm the convergence analysis. Adetailed comparison with several popular time stepping schemes is alsoperformed. The numerical results indicate that the proposed fully discreteschemes are accurate and robust for nonsmooth data, and competitive withexisting schemes.

Two Schemes for Fractional Diffusion and Diffusion-Wave Equations with Nonsmooth Data