We present a cut finite element method for the heat equation on two overlapping meshes: a stationary background mesh and an overlapping mesh that evolves inside/"on top" of it. Here the overlapping mesh is prescribed a simple discontinuous evolution, meaning that its location, size, and shape as functions of time are discontinuous and piecewise constant. 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. The simple discontinuous mesh evolution results in a space-time discretization with a slabwise product structure between space and time which allows for existing analysis methodologies to be applied with only minor modifications. We follow the analysis methodology presented by Eriksson and Johnson in [1, 2]. The greatest modification is the introduction of a Ritzlike "shift operator" that is used to obtain the discrete strong stability needed for the error analysis. The shift operator generalizes the original analysis to some methods for which the discrete subspace at one time does not lie in the space of the stiffness form at the subsequent time. The error analysis consists of 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剪切的限定元素方法:固定背景网格和重叠网格,在它内部/“顶部”中演化。在这里,重叠网格被指定为简单的不连续进化,这意味着它的位置、大小和形状作为时间函数的形状不连续和片度常态。对于离散功能空间,我们在空间和不连续的Galerkin使用连续的Galerkin 和不连续的Galerkin 方法,同时在两个梅什之间的边界上添加不连续的连接。有限的元素配方以尼察的方法为基础。简单的不连续网格进化使时空时的产品结构离散化,使现有的分析方法在空间和时间之间只能进行细小的修改。我们遵循埃里克松和约翰逊在[1、2] 中提出的分析方法。最大的修改是引入一个类似“变换操作器”的“变换操作器”,用来获得错误分析所需的离散强的稳定性。变换操作员将原始分析结果概括为某些方法,在某个时空子空间的离散子空间的分空间,一个时空段的变的变法则由一个最精确的错误构成一个最精确的校差的校正。一个空间的校正的校正,一个空间的校正的校正的校正。</s>