  In this work, we study the numerical approximation of local fluctuations of
certain classes of parabolic stochastic partial differential equations (SPDEs).
Our focus is on effects for small spatially-correlated noise on a time scale
before large deviation effects have occurred. In particular, we are interested
in the local directions of the noise described by a covariance operator. We
introduce a new strategy and prove a Combined ERror EStimate (CERES) for the
five main errors: the spatial discretization error, the local linearization
error, the noise truncation error, the local relaxation error to steady state,
and the approximation error via an iterative low-rank matrix algorithm. In
summary, we obtain one CERES describing, apart from modelling of the original
equations and standard round-off, all sources of error for a local fluctuation
analysis of an SPDE in one estimate. To prove our results, we rely on a
combination of methods from optimal Galerkin approximation of SPDEs, covariance
moment estimates, analytical techniques for Lyapunov equations, iterative
numerical schemes for low-rank solution of Lyapunov equations, and working with
related spectral norms for different classes of operators.
