In the context of coupling methods that match molecular dynamics with finite elements, we focus on the finite temperature and heat rates treatment done by Bridging Domain approaches. The energy dissipation occurring due to the projection of the fine scale onto the coarse scale is demonstrated during a sliding simulation of contacting surfaces. We describe the problems brought by the dissipation and propose a method that does not alter the temperature of the atoms in the coupling region. A coupling of the thermal fluxes with a continuum temperature field is also introduced and tested on simple examples.