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This article is concerned with the discretisation of the Stokes equations ontime-dependent domains in an Eulerian coordinate framework. Our work can beseen as an extension of a recent paper by Lehrenfeld & Olshanskii [ESAIM: M2AN,53(2):585-614, 2019], where BDF-type time-stepping schemes are studied for aparabolic equation on moving domains. For space discretisation, a geometricallyunfitted finite element discretisation is applied in combination with Nitsche'smethod to impose boundary conditions. Physically undefined values of thesolution at previous time-steps are extended implicitly by means of so-calledghost penalty stabilisations. We derive a complete a priori error analysis ofthe discretisation error in space and time, including optimal $L^2(L^2)$-normerror bounds for the velocities. Finally, the theoretical results aresubstantiated with numerical examples.

Eulerian time-stepping schemes for the non-stationary Stokes equations on time-dependent domains