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arXiv · 1910.03054

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

Erik Burman·Stefan Frei·Andre Massing
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Abstract

This article is concerned with the discretisation of the Stokes equations on time-dependent domains in an Eulerian coordinate framework. Our work can be seen 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 a parabolic equation on moving domains. For space discretisation, a geometrically unfitted finite element discretisation is applied in combination with Nitsche's method to impose boundary conditions. Physically undefined values of the solution at previous time-steps are extended implicitly by means of so-called ghost penalty stabilisations. We derive a complete a priori error analysis of the discretisation error in space and time, including optimal $L^2(L^2)$-norm error bounds for the velocities. Finally, the theoretical results are substantiated with numerical examples.

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Erik Burman, Stefan Frei, Andre Massing. 2019-10-07. Eulerian time-stepping schemes for the non-stationary Stokes equations on time-dependent domains. https://arxiv.org/abs/1910.03054

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