  This paper is concerned with the adaptive numerical treatment of stochastic
partial differential equations. Our method of choice is Rothe's method. We use
the implicit Euler scheme for the time discretization. Consequently, in each
step, an elliptic equation with random right-hand side has to be solved. In
practice, this cannot be performed exactly, so that efficient numerical methods
are needed. Well-established adaptive wavelet or finite-element schemes, which
are guaranteed to converge with optimal order, suggest themselves. We
investigate how the errors corresponding to the adaptive spatial discretization
propagate in time, and we show how in each time step the tolerances have to be
chosen such that the resulting perturbed discretization scheme realizes the
same order of convergence as the one with exact evaluations of the elliptic
subproblems.
