  A high-order finite element method is proposed to solve the nonlinear
convection-diffusion equation on a time-varying domain whose boundary is
implicitly driven by the solution of the equation. The method is semi-implicit
in the sense that the boundary is traced explicitly with a high-order
surface-tracking algorithm, while the convection-diffusion equation is solved
implicitly with high-order backward differentiation formulas and
fictitious-domain finite element methods. By two numerical experiments for
severely deforming domains, we show that optimal convergence orders are
obtained in energy norm for third-order and fourth-order methods.
