Matrix-Free Ghost Penalty Evaluation via Tensor Product Factorization
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915829282504704 |
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| author | Wichrowski, Michał |
| author_facet | Wichrowski, Michał |
| contents | 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 |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_00246 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Matrix-Free Ghost Penalty Evaluation via Tensor Product Factorization Wichrowski, Michał Numerical Analysis 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 |
| title | Matrix-Free Ghost Penalty Evaluation via Tensor Product Factorization |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2503.00246 |