  We study solution techniques for parabolic equations with fractional
diffusion and Caputo fractional time derivative, the latter being discretized
and analyzed in a general Hilbert space setting. The spatial fractional
diffusion is realized as the Dirichlet-to-Neumann map for a nonuniformly
elliptic problem posed on a semi-infinite cylinder in one more spatial
dimension. We write our evolution problem as a quasi-stationary elliptic
problem with a dynamic boundary condition. We propose and analyze an implicit
fully-discrete scheme: first-degree tensor product finite elements in space and
an implicit finite difference discretization in time. We prove stability and
error estimates for this scheme.
