Time-stepping a PDE on a moving domain with a fixed background mesh fails naively because a finite element function on one step's active mesh is undefined on the next step's, leaving the time difference quotient meaningless. The remedy is to solve instead for a Sobolev extension of the solution, which is legitimate because continuous extension operators exist and commute with the time derivative - and, decisively, never needs to be computed: adding a ghost-penalty stabilization term over a narrow band around the moving boundary makes a suitable discrete extension appear on its own. This buys a scheme built only from standard stationary unfitted finite elements and standard finite differences, needing no space-time domain reconstruction, no reference-domain mapping, and no CFL-type time step restriction, with unconditional energy stability and an optimal-order error bound separating temporal, geometric and spatial contributions.
Christoph Lehrenfeld (Institute for Numerical and Applied Mathematics, University of Gottingen) and Maxim A. Olshanskii (University of Houston), arXiv 1803.01779v2, math.NA, August 2018, twenty-seven…