  We apply a space adaptive interior penalty discontinuous Galerkin method for
solving advective Allen-Cahn equation with expanding and contracting velocity
fields. The advective Allen-Cahn equation is first discretized in time and the
resulting semi-linear elliptic PDE is solved by an adaptive algorithm using a
residual-based a posteriori error estimator. The a posteriori error estimator
contains additional terms due to the non-divergence-free velocity field.
Numerical examples demonstrate the effectiveness and accuracy of the adaptive
approach by resolving the sharp layers accurately.
