Isoparametric unfitted BDF -- Finite element method for PDEs on evolving domains

We propose a new discretization method for PDEs on moving domains in the setting of unfitted finite element methods, which is provably higher-order accurate in space and time. In the considered setting, the physical domain that evolves essentially arbitrarily through a time-independent computational background domain, is represented by a level set function. For the time discretization, the application of standard time stepping schemes that are based on finite difference approximations of the time derivative is not directly possible, as the degrees of freedom may get active or inactive across such a finite difference stencil in time. In [Lehrenfeld, Olshanskii. An Eulerian finite element method for PDEs in time-dependent domains. ESAIM: M2AN, 53:585--614, 2019] this problem is overcome by extending the discrete solution at every timestep to a sufficiently large neighborhood so that all the degrees of freedom that are relevant at the next time step stay active. But that paper focuses on low-order methods. We advance these results with introducing and analyzing realizable techniques for the extension to higher order. To obtain higher-order convergence in space and time, we combine the BDF time stepping with the isoparametric unfitted FEM. The latter has been used and analyzed for several stationary problems before. However, for moving domains the key ingredient in the method, the transformation of the underlying mesh, becomes time-dependent which gives rise to some technical issues. We treat these with special care, carry out an a priori error analysis and two numerical experiments.