  We present a strategy for the numerical solution of convection-coupled
phase-transition problems, with focus on solidification and melting. We solve
for the temperature and flow fields over time. The position of the phase-change
interface is tracked with a level-set method, which requires knowledge of the
heat-flux discontinuity at the interface. In order to compute the heat-flux
jump, we build upon the ghost-cell approach and extend it to the space-time
finite element method. This technique does not require a local enrichment of
the basis functions, such as methods like extended finite elements, and it can
be easily implemented in already existing finite element codes. Verification
cases for the 1D Stefan problem and the lid-driven cavity melting problem are
provided. Furthermore, we show a more elaborate 2D case in view of complex
applications.
