  Explicit stabilized integrators are an efficient alternative to implicit or
semi-implicit methods to avoid the severe timestep restriction faced by
standard explicit integrators applied to stiff diffusion problems. In this
paper, we provide a fully discrete strong convergence analysis of a family of
explicit stabilized methods coupled with finite element methods for a class of
parabolic semilinear deterministic and stochastic partial differential
equations. Numerical experiments including the semilinear stochastic heat
equation with space-time white noise confirm the theoretical findings.
