  A Cahn-Hilliard-Allen-Cahn phase-field model coupled with a heat transfer
equation, particularly with full non-diagonal mobility matrices, is studied.
After reformulating the problem w.r.t. the inverse of temperature, we proposed
and analysed a structure-preserving approximation for the semi-discretisation
in space and then a fully discrete approximation using conforming finite
elements and time-stepping methods. We prove structure-preserving property and
discrete stability using relative entropy methods for the semi-discrete and
fully discrete case. The theoretical results are illustrated by numerical
experiments.
