  Existence and uniqueness for semilinear stochastic evolution equations with
additive noise by means of finite dimensional Galerkin approximations is
established and the convergence rate of the Galerkin approximations to the
solution of the stochastic evolution equation is estimated. These abstract
results are applied to several examples of stochastic partial differential
equations (SPDEs) of evolutionary type including a stochastic heat equation, a
stochastic reaction diffusion equation and a stochastic Burgers equation. The
estimated convergence rates are illustrated by numerical simulations. The main
novelty in this article is to estimate the difference of the finite dimensional
Galerkin approximations and of the solution of the infinite dimensional SPDE
uniformly in space, i.e., in the L^{\infty}-topology, instead of the usual
Hilbert space estimates in the L^2-topology, that were shown before.
