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This paper is concerned with the adaptive numerical treatment of stochasticpartial differential equations. Our method of choice is Rothe's method. We usethe implicit Euler scheme for the time discretization. Consequently, in eachstep, an elliptic equation with random right-hand side has to be solved. Inpractice, this cannot be performed exactly, so that efficient numerical methodsare needed. Well-established adaptive wavelet or finite-element schemes, whichare guaranteed to converge with optimal order, suggest themselves. Weinvestigate how the errors corresponding to the adaptive spatial discretizationpropagate in time, and we show how in each time step the tolerances have to bechosen such that the resulting perturbed discretization scheme realizes thesame order of convergence as the one with exact evaluations of the ellipticsubproblems.

On the convergence analysis of the inexact linearly implicit Euler scheme for a class of SPDEs