  An initial-boundary value problem for the time-fractional diffusion equation
is discretized in space using continuous piecewise-linear finite elements on a
polygonal domain with a re-entrant corner. Known error bounds for the case of a
convex polygon break down because the associated Poisson equation is no longer
$H^2$-regular. In particular, the method is no longer second-order accurate if
quasi-uniform triangulations are used. We prove that a suitable local mesh
refinement about the re-entrant corner restores second-order convergence. In
this way, we generalize known results for the classical heat equation due to
Chatzipantelidis, Lazarov, Thom\'ee and Wahlbin.
