  We numerically investigate the spatial and temporal statistical properties of
a dilute polymer solution in elastic turbulence regime, i.e., in the chaotic
flow state occurring at vanishing Reynolds and high Weissenberg numbers. We aim
at elucidating the relations between measurements of flow properties performed
in the spatial domain with the ones taken in the temporal domain, which is a
key point for the interpretation of experimental results on elastic turbulence.
Indeed, the available experimental techniques typically allow access to the
velocity field at best in a few specific positions and at various times so that
one can retrieve the spatial flow statistics only through Taylor's hypothesis.
To this end, we carry out extensive direct numerical simulations of the
two-dimensional Kolmogorov flow of an Oldroyd-B viscoelastic fluid. Static
point-like numerical probes are placed at different locations in the flow,
particularly at the extrema of mean flow amplitude. The results in the fully
developed elastic turbulence regime reveal large velocity fluctuations, as
compared to the mean flow, leading to a partial breakdown of Taylor's
frozen-field hypothesis. While second-order statistics, probed by spectra and
structure functions, display consistent scaling behaviors in the spatial and
temporal domains, the third-order statistics highlight robust differences. In
particular, the temporal analysis fails to capture the skewness of streamwise
longitudinal velocity increments. Finally, we assess both the degree of
statistical inhomogeneity and isotropy of the flow turbulent fluctuations as a
function of scale. While the system is only weakly non-homogenous in the
cross-stream direction, it is found to be highly anisotropic at all scales.
