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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. ThesearXiv:1404.3800v4 fatcat:dtrrca3slzcbvigrzcofb55qpi