  Immersed finite element (IFE) methods are a group of long-existing numerical
methods for solving interface problems on unfitted meshes. A core argument of
the methods is to avoid mesh regeneration procedure when solving moving
interface problems. Despite the various applications in moving interface
problems, a complete theoretical study on the convergence behavior is still
missing. This research is devoted to close the gap between numerical
experiments and theory. We present the first fully discrete analysis including
the stability and optimal error estimates for a backward Euler IFE method for
solving parabolic moving interface problems. Numerical results are also
presented to validate the analysis.
