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). 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 method is implemented within the \texttt{deal.II} library.