Matrix-Free Ghost Penalty Evaluation via Tensor Product Factorization

Abstract

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. The source code used for this paper is available at https://github.com/mwichro/TensorGhostPenalty