  The hot gas embedded in the large-scale structures in the Universe produces
secondary fluctuations in the Cosmic Microwave Background (CMB). Because it is
proportional to the gas pressure integrated along the line of sight, this
effect, the thermal Sunyaev-Zel'dovich (SZ) effect, provides a direct measure
of large-scale structure and of cosmological parameters. We study the
statistical properties of this effect using both hydrodynamical simulations and
analytical predictions from an extended halo model. The Adaptive Mesh
Refinement scheme, used in the newly developed code RAMSES, provides a dynamic
range of 4 order of magnitudes, and thus allows us to significantly improve
upon earlier calculations. After accounting for the finite mass resolution and
box size of the simulation, we find that the halo model agrees well with the
simulations. We discuss and quantify the uncertainty in both methods, and thus
derive an accurate prediction for the SZ power spectrum in the 10^2<l<10^5
range of multipole. We show how this combined analytical and numerical approach
is essential for accuracy, and useful for the understanding of the physical
processes and scales which contribute to large-scale SZ anisotropies.
