Isoparametric Unfitted BDF--Finite Element Method for PDEs on Evolving Domains

Abstract

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 become active or inactive across such a finite difference stencil in time. In [C. Lehrenfeld and M. A. Olshanskii, ESAIM Math. Model. Numer. Anal., 53 (2019), pp. 585--614], this problem is overcome by extending the discrete solution at every time step to a sufficiently large neighborhood so that all 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 by introducing and analyzing realizable techniques for the extension to higher order. To obtain higher-order convergence in space and time, we combine backward differentiation formula (BDF) time stepping with the isoparametric unfitted finite element method (FEM). The latter has been used and analyzed for several stationary problems. 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 present two numerical experiments.