The evolution of gas bubbles on solid surfaces is known to affect the efficiency of various heat transfer and electrolysis processes. In this work, the over-saturation-driven growth of air bubbles nucleating at micro-cavities is investigated by direct numerical simulations, and the results are directly compared with experimental results obtained at silicon wafers of different wettability and size of the micro-cavities. A validated geometric volume-of-fluid method with adaptive mesh refinement is used to accurately capture the mass transfer at the interface. Applying proper boundary conditions then allows to accurately reproduce the two-stage wetting dynamics observed in experiments, where the contact line of the growing bubble initially remains pinned at the cavity edge before it later starts to move with constant contact angle. The bubble growth behavior obtained numerically is further compared and discussed with results from analytical approaches of an extended Scriven and the Popov solution, which give accurate results only at idealized initial conditions and low values of the oversaturation, where convective effects can be neglected. The simulation results obtained at surfaces of different wettability and cavity size allow to gain insight how the bubble behavior can be controlled by functionalization and structuring of the surface. The numerical approach presented in this work may easily be applied to more complex situations where theoretical solutions do not hold or experiments are not feasible.