  We consider the initial/boundary value problem for the fractional diffusion
and diffusion-wave equations involving a Caputo fractional derivative in time.
We develop two "simple" fully discrete schemes based on the Galerkin finite
element method in space and convolution quadrature in time with the generating
function given by the implicit backward Euler method/second-order backward
difference method, and establish error estimates optimal with respect to the
regularity of the initial data. These two schemes are first and second-order
accurate in time for nonsmooth initial data. Extensive numerical experiments
for one and two-dimensional problems confirm the convergence analysis. A
detailed comparison with several popular time stepping schemes is also
performed. The numerical results indicate that the proposed fully discrete
schemes are accurate and robust for nonsmooth data, and competitive with
existing schemes.
