We present a matrix-free approach for implementing ghost penalty stabilization in Cut Finite Element Methods (CutFEM). While matrix-free methods for CutFEM have been developed, the efficient evaluation of high-order, face-based ghost penalties remains a significant challenge, which this work addresses. By exploiting the tensor-product structure of the ghost penalty operator, we reduce its evaluation to a series of one-dimensional matrix-vector products using precomputed 1D matrices, avoiding the need to evaluate high-order derivatives directly. This approach achieves $O(k^{d+1})$ complexity for elements of degree $k$ in $d$ dimensions, significantly reducing implementation effort while maintaining accuracy. The derivation relies on the fact that the cells are aligned with the coordinate axes. The method is implemented within the \texttt{deal.II} library.

翻译：本文提出了一种在切割有限元方法（CutFEM）中实现鬼影罚项稳定的无矩阵方法。尽管针对CutFEM的无矩阵方法已有研究，但高阶、基于面的鬼影罚项的高效评估仍是一个重要挑战，本工作正是为了解决这一问题。通过利用鬼影罚项算子的张量积结构，我们将其评估转化为使用预计算一维矩阵进行的一系列一维矩阵-向量乘积，从而避免直接计算高阶导数。该方法在$d$维空间中对于阶数为$k$的单元实现了$O(k^{d+1})$的计算复杂度，在保持精度的同时显著降低了实现难度。该推导依赖于单元与坐标轴对齐的前提条件。本方法已在\\texttt{deal.II}库中实现。