The authors consider the pathwise numerical approximation of the stochastic evolution equation
on the Hilbert space , where is the generator of an analytic semigroup , , is an -valued -Wiener process, the mapping is nonlinear with bounded first and second derivatives. They also assume that and have common eigenfunctions. However, the covariance operator needs not to be a trace-class operator.
To treat this equation numerically, the authors discretise in space by a Galerkin method and in time by a stochastic exponential integrator. They estimate the rates of convergence for both th-mean and pathwise errors of approximations. It appears that for spatially regular noise the number of nodes needed for the noise can be reduced and that the rate of convergence degrades as the regularity of the noise reduces. The results are illustrated by the two-dimensional Allen–Cahn equation.