  Parameter estimation for a parabolic linear stochastic partial differential
equation in one space dimension is studied observing the solution field on a
discrete grid in a fixed bounded domain. Considering an infill asymptotic
regime in both coordinates, we prove central limit theorems for realized
quadratic variations based on temporal and spatial increments as well as on
double increments in time and space. Resulting method of moments estimators for
the diffusivity and the volatility parameter inherit the asymptotic normality
and can be constructed robustly with respect to the sampling frequencies in
time and space. Upper and lower bounds reveal that in general the optimal
convergence rate for joint estimation of the parameters is slower than the
usual parametric rate. The theoretical results are illustrated in a numerical
example.
