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We propose a new discretization method for PDEs on moving domains in thesetting of unfitted finite element methods, which is provably higher-orderaccurate in space and time. In the considered setting, the physical domain thatevolves essentially arbitrarily through a time-independent computationalbackground domain, is represented by a level set function. For the timediscretization, the application of standard time stepping schemes that arebased on finite difference approximations of the time derivative is notdirectly possible, as the degrees of freedom may get active or inactive acrosssuch a finite difference stencil in time. In [Lehrenfeld, Olshanskii. AnEulerian finite element method for PDEs in time-dependent domains. ESAIM: M2AN,53:585--614, 2019] this problem is overcome by extending the discrete solutionat every timestep to a sufficiently large neighborhood so that all the degreesof freedom that are relevant at the next time step stay active. But that paperfocuses on low-order methods. We advance these results with introducing andanalyzing realizable techniques for the extension to higher order. To obtainhigher-order convergence in space and time, we combine the BDF time steppingwith the isoparametric unfitted FEM. The latter has been used and analyzed forseveral stationary problems before. However, for moving domains the keyingredient in the method, the transformation of the underlying mesh, becomestime-dependent which gives rise to some technical issues. We treat these withspecial care, carry out an a priori error analysis and two numericalexperiments.

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