We study the melting dynamics of a phase change material by the action of thermocapillary driving. The geometry corresponds to squares and rectangles with a top free flat surface. We simulate a phase change material with Prandtl number Pr~ 60 in the liquid phase and two Stefan numbers S t e= 0.67 and S t e= 0.33. The interplay between conduction and convection dominated regions within the liquid phase of the PCM determines the melting dynamics. We find that some magnitudes follow simple power laws; such as the position of the melting front, the time to full melting, and the input of energy into the system. In addition, we find that the dynamic Marangoni number scales with the Nusselt number as a power-law Nu~ Ma α, where the exponent α depends on the aspect ratio of the geometry and position of the melting fronts at the surface and bottom. These scalings prevail in a wide range of Marangoni numbers and are only slightly affected by the Stefan number.