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：我们为两个重叠的模层的热方程提出了一个削减的有限元素方法:固定背景网和重叠网,在其中/“顶部”周围移动。这里,重叠网格被指定为简单的连续运动,这意味着它作为时间函数的位置是连续的和片向线的。对于离散的功能空间,我们使用空间中的连续Galerkin和不连续的Galerkin,同时在两个梅什之间的边界上添加一个不连续的参数。有限元素的配方以尼采的方法为基础,还包括两个模拟标准不连续的Galerkin时间-jump 术语之间的时空边界的一个整体术语。简单的连续网格运动导致空间时分化,而标准的分析方法要么失败,要么不合适。因此,我们使用一个似乎较新的能源分析框架,这个框架很一般,而且坚固,足以适用于当前环境。能源分析包含一种比标准基本方法略强一点的稳定性估计,以及一个先有误差的时空边界,即模拟两个模,即模拟标准不连续的Galerkin-cal-maimal laimal laim laim laim a pal laim pal lagime laim pal a pal pal a pal lagime lagild lagild lax lagime ladal lax lax lax lax lax lax lautal a latical latial lati lati latial lati lati latial latial lati lax lax lax lax lax lax lax lax lax lax lax 和 lax lax 和 lax lax labal lax 和 和 lax lax lax lax lax 和 lax 和 和 latial latical lax lax lax lax lax lax lax lax lax lax latial latial labal lati lati lati lax latial la</s>