Standard models for a warm, wet early Mars require a significant CO2-H2O atmosphere in the past. The source for these phases is assumed to be volcanic degassing. However, no consistent, dynamical models exist relating volcanic degassing to evolving mantle temperatures. Here we use a range of thermal, geophysical, geological, and petrological constraints from Mars to constrain mantle convection model simulations of Mars' post-Noachian stagnant lid evolution. We develop a methodology to self-consistently calculate melt extraction from the mantle source region. Using a dike-propagation algorithm, we can calculate the rate of volcanism and rate of volcanic degassing from these simulations and compare them with estimates for Mars. We find that Martian melt production rates are satisfied by a 200-km thick lithosphere (surface heat flow 25 ± 5 mW/m3) for an intermediate Martian solidus. Core-mantle temperatures cannot exceed ∼1850°C from geodynamo constraints, and the enrichment of heat-producing elements into the crust is unlikely to exceed 25-50%. For hotter Martian mantle temperatures in the past, we find an evolution in rates of volcanism from >0.17 km3/yr for the early Hesperian to ∼1 × 10-4 km3 /yr at present, consistent with geological evidence. During this same interval, CO2 flux would have declined from 8.8 × 107 to 6.7 × 106 kg/yr. If the early Hesperian supported a dense (>1 bar) atmosphere, this implies that the average loss rate of CO2 from the atmosphere was 15 times greater than the maximum influx rate during this time.