We consider a time-stepping scheme of Crank-Nicolson type for the heat equation on a moving domain in Eulerian coordinates. As the spatial domain varies between subsequent time steps, an extension of the solution at the previous time step is required. Following Lehrenfeld \& Olskanskii [ESAIM: M2AN, 53(2):\,585-614, 2019], we apply an implicit extension based on so-called ghost-penalty terms. For spatial discretisation, a cut finite element method is used. We derive a complete a priori error analysis in space and time, which shows in particular second-order convergence in time under a parabolic CFL condition. Finally, we present numerical results in two and three space dimensions that confirm the analytical estimates.
翻译：我们考虑对欧莱安坐标移动域域的热等式采用Crank-Nicolson型的时序计划。 由于空间域在随后的时序中各不相同, 需要在前一个时序中延长解决方案。 在Lehrenfeld ⁇ Olskanskii [ESAIM: M2AN, 53(2):\, 585-614, 2019] [ESAIM: M2AN, 53(2):\, 585-614, 2019] 之后, 我们根据所谓的幽灵- 刑罚条件, 使用隐含的延长。 对于空间离异化, 使用一个削减的有限元素方法。 我们在时空上进行完全的先验误差分析, 这表明在抛光的 CFL 条件下, 特别是第二级的合并。 最后, 我们用两个和三个空间层面的数值结果来证实分析估计。
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