Numerical simulations of two-dimensional Rayleigh-Bénard convection are designed to study lithospheric cooling above a convective mantle. A strongly temperature-and pressure-dependent viscosity fluid is heated from below or from within. An imposed velocity at the surface of the box mimicks the plate motion between the ridge on one side and the subduction zone on the other side. As the lithosphere cools, its upper part remains rigid and therefore conductive, while its bottom part is convectively unstable. Dripping instabilities are not observed close to the ridge. Nevertheless, the material flows along the slope defined by the lower part of the lithosphere and feeds the first descending drip. Afterwards, cold downgoing instabilities develop continuously and randomly at the base of the lithosphere and are replaced by hot material from the convecting core of the box. The lithosphere continues to thicken even after the onset of the first instability. Surface heat flow, subsidence and lithospheric temperature structure obtained by the convective simulations are compared to the predictions of three conductive models: the Plate, Chablis, and modified Chablis models. These models differ by their applied bottom boundary condition which represents the lithosphere/asthenosphere convective coupling, i.e. by the presence or absence of instabilities developing at the base of the lithosphere. The conductive model which best explains the lithospheric cooling obtained by convective simulations is the modified Chablis model. In this model, a variable heat flow (depending upon the viscosity at the base of the lithosphere) is applied along an isotherm located in the lower unstable part of the lithosphere.