We present a cut finite element method for the heat equation on two overlapping meshes. By overlapping meshes we mean a mesh hierarchy with a stationary background mesh at the bottom and an overlapping mesh that is allowed to move around on top of the background mesh. Overlapping meshes can be used as an alternative to costly remeshing for problems with changing or evolving interior geometry. In this paper the overlapping mesh is prescribed a cG(1) movement, meaning that its location as a function of time is continuous and piecewise linear. For the discrete function space, we use continuous Galerkin in space and discontinuous Galerkin in time, with the addition of a discontinuity on the boundary between the two meshes. The finite element formulation is based on Nitsche's method and also includes an integral term over the space-time boundary that mimics the standard discontinuous Galerkin time-jump term. The cG(1) mesh movement results in a space-time discretization for which existing analysis methodologies either fail or are unsuitable. We therefore propose, to the best of our knowledge, a new energy analysis framework that is general and robust enough to be applicable to the current setting$^*$. The energy analysis consists of a stability estimate that is slightly stronger than the standard basic one and an a priori error estimate that is of optimal order with respect to both time step and mesh size. We also present numerical results for a problem in one spatial dimension that verify the analytic error convergence orders. $*$ UPDATE and CORRECTION: After this work was made public, it was discovered that the core components of the new energy analysis framework seemed to have been discovered independently by us and Cangiani, Dong, and Georgoulis in [1].

翻译：我们提出一种用于两个重叠网格上热方程的切割有限元方法。所谓重叠网格，是指一种网格层级结构，其底部包含一个固定的背景网格，顶部则允许一个重叠网格在背景网格上移动。重叠网格可作为针对具有变化或演化内部几何形状问题的高代价重新网格化的替代方案。本文中，重叠网格被指定为cG(1)运动，即其位置随时间连续且分段线性。对于离散函数空间，我们采用空间连续伽辽金法和时间间断伽辽金法，并在两个网格之间的边界上附加一个间断项。有限元公式基于尼采法，并包含一个时空边界上的积分项，该积分项模拟了标准间断伽辽金时间跳跃项。cG(1)网格运动导致了一种时空离散化，对于这种离散化，现有的分析方法要么失效，要么不合适。因此，据我们所知，我们提出了一种新的能量分析框架，该框架足够通用且稳健，适用于当前设定$^*$。该能量分析包含一个略强于标准基本估计的稳定性估计，以及一个关于时间步长和网格大小均为最优阶的先验误差估计。我们还给出了一个一维空间问题的数值结果，验证了解析误差的收敛阶数。$*$ 更新与修正：本工作公开后，我们发现该新能量分析框架的核心组成部分似乎已被我们以及Cangiani、Dong和Georgoulis在[1]中独立发现。