We present a cut finite element method for the heat equation on two overlapping meshes: a stationary background mesh and an overlapping mesh that moves around inside/"on top" of it. Here the overlapping mesh is prescribed a simple continuous motion, 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 between the two meshes that mimics the standard discontinuous Galerkin time-jump term. The simple continuous mesh motion results in a space-time discretization for which standard analysis methodologies either fail or are unsuitable. We therefore employ what seems to be a relatively 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.

翻译：本文针对两个重叠网格上的热方程提出一种切割有限元方法：一个固定背景网格和一个在其内部/"上方"运动的重叠网格。该重叠网格被赋予简单的连续运动，即其位置作为时间函数是连续且分段线性的。对于离散函数空间，我们在空间上采用连续伽辽金法，时间上采用间断伽辽金法，并在两个网格边界处引入不连续性。有限元公式基于尼采方法，并在两个网格的时空边界上增加一个积分项，以模拟标准间断伽辽金时间跳跃项。简单的连续网格运动导致时空离散化，标准分析方法对此要么失效要么不适用。因此，我们采用一种相对较新的能量分析框架，该框架通用且稳健，足以适用于当前场景。能量分析包括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尺寸均为最优阶的先验误差估计。我们还在一个空间维度问题上给出了数值结果，验证了解析误差收敛阶。