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We consider initial/boundary value problems for the subdiffusion and diffusionwave equations involving a Caputo fractional derivative in time. We develop two fully discrete schemes based on the piecewise linear Galerkin finite element method in space and convolution quadrature in time with the generating function given by the backward Euler method/second-order backward difference method, and establish error estimates optimal with respect to the regularity of problem data. These two schemes aredoi:10.1137/140979563 fatcat:odijwwnzavbtvcykhth25hp6qq