  Using a continuum bead-spring Monte Carlo model, we study the anomalous
diffusion dynamics of a self-avoiding tethered membrane by means of extensive
computer simulations. We focus on the subdiffusive stochastic motion of the
membrane's central node in the regime of flat membranes at temperatures above
the membrane folding transition. While at times, larger than the characteristic
membrane relaxation time $\tau_R$, the mean-square displacement of the center
of mass of the sheet, $<R_c^2>$, as well as that of its central node,
$<R_n^2>$, show the normal Rouse diffusive behavior with a diffusion
coefficient $D_N$ scaling as $D_N \propto N^{-1}$ with respect to the number of
segments $N$ in the membrane, for short times $t\le \tau_R$ we observe a {\em
multiscale dynamics} of the central node, $<R_n^2> \propto t^\alpha$, where the
anomalous diffusion exponent $\alpha$ changes from $\alpha \approx 0.86$ to
$\alpha \approx 0.27$, and then to $\alpha \approx 0.5$, before diffusion turns
eventually to normal. By means of simple scaling arguments we show that our
main result, $\alpha \approx 0.27$, can be related to particular mechanisms of
membrane dynamics which involve different groups of segments in the membrane
sheet. A comparative study involving also linear polymers demonstrates that the
diffusion coefficient of self-avoiding tethered membranes, containing $N$
segments, is three times smaller than that of linear polymer chains with the
same number of segments.
