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：我们提出了一种面向热传导方程的切割有限元方法，该方法基于两套重叠网格：一套固定的背景网格和一套在其内部/上方演化的重叠网格。其中重叠网格被赋予简单的非连续演化模式，即其位置、尺寸和形状随时间呈非连续且逐段常数的变化。离散函数空间采用空间连续伽辽金法与时间间断伽辽金法的组合，并在两套网格的边界处引入不连续性。有限元公式基于尼采方法。这种简单的非连续网格演化形成具有时空切片乘积结构的时空离散格式，使得现有分析方法仅需微小调整即可应用。我们遵循埃里克森和约翰逊在[1,2]中提出的分析框架。最主要的改进是引入类似里茨的"位移算子"，用于获取误差分析所需的离散强稳定性。该算子将原始分析推广至某些方法——在这些方法中，某时刻的离散子空间并不属于后续时刻刚度形式所在的函数空间。误差分析包含针对时间步长和网格尺寸均达到最优阶的先验误差估计。我们还呈现了空间一维问题的数值结果，验证了解析误差收敛阶。