  We study a semilinear fractional-in-time Rayleigh-Stokes problem for a
generalized second-grade fluid with a Lipschitz continuous nonlinear source
term and initial data $u_0\in\dot{H}^\nu(\Omega)$, $\nu\in[0,2]$. We discuss
stability of solutions and provide regularity results. Two spatially
semidiscrete schemes are analyzed based on standard Galerkin and lumped mass
finite element methods, respectively. Further, a fully discrete scheme is
obtained by applying a convolution quadrature in time generated by the backward
Euler method, and optimal error estimates are derived for smooth and nonsmooth
initial data. Finally, numerical examples are provided to illustrate the
theoretical results.
