  We consider the semilinear stochastic heat equation perturbed by additive
noise. After time-discretization by Euler's method the equation is split into a
linear stochastic equation and a non-linear random evolution equation. The
linear stochastic equation is discretized in space by a non-adaptive
wavelet-Galerkin method. This equation is solved first and its solution is
substituted into the nonlinear random evolution equation, which is solved by an
adaptive wavelet method. We provide mean square estimates for the overall
error.
