  In this study, we employ analytical and numerical techniques to examine a
phase transition model with moving boundaries. The model displays two relevant
spatial scales pointing out to a macroscopic phase and a microscopic phase,
interacting on disjoint inclusions. The shrinkage or the growth of the
inclusions is governed by a modified Gibbs-Thomson law depending on the
macroscopic temperature, but without accessing curvature information. We use
the Hanzawa transformation to transform the problem onto a fixed reference
domain. Then a fixed-point argument is employed to demonstrate the
well-posedness of the system for a finite time interval. Due to the model's
nonlinearities and the macroscopic parameters, which are given by differential
equations that depend on the size of the inclusions, the problem is
computationally expensive to solve numerically. We introduce a precomputing
approach that solves multiple cell problems in an offline phase and uses an
interpolation scheme afterward to determine the needed parameters.
Additionally, we propose a semi-implicit time-stepping method to resolve the
nonlinearity of the problem. We investigate the errors of both the precomputing
and time-stepping procedures and verify the theoretical results via numerical
simulations.
