We investigate the geodynamic and melting history of Mars using 3D spherical shell models of mantle convection, constrained by the recent InSight mission results. The Martian mantle must have produced sufficient melt to emplace the Tharsis rise by the end of the Noachian–requiring on the order of 1–3×109 km3 of melt after accounting for limited (~10\%) melt extraction. Thereafter, melting declined, but abundant evidence for limited geologically recent volcanism necessitates some melt even in the cool present-day mantle inferred from InSight data. We test models with two mantle activation energies, and a range of crustal Heat Producing Element (HPE) enrichment factors and initial core-mantle boundary temperatures. We also test the effect of including a hemispheric (spherical harmonic degree-1) step in lithospheric thickness to model the Martian dichotomy. We find that a higher activation energy (350 kJ mol−1) rheology produces present-day geotherms consistent with InSight results, and of those the cases with HPE enrichment factors of 5–10x produce localized melting near or up to present-day. 10x crustal enrichment is consistent with both InSight and geochemical results, and those models also produce present-day geoid power spectra consistent with Mars. However, it is very difficult to produce sufficient melt to form Tharsis in a mantle that also matches the present-day geotherm, without assuming extremely efficient extraction of melt to the surface. The addition of a degree-1 hemispheric dichotomy, as an equatorial step in lithospheric thickness, does not significantly improve upon melt production or the geoid.