Multigrid p-Robustness at Jacobi Speeds: Efficient Matrix-Free Implementation of Local p-Multigrid Solvers
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911297931575296 |
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| author | Wichrowski, Michał |
| author_facet | Wichrowski, Michał |
| contents | Vertex-patch smoothers are essential for the robust convergence of geometric multigrid methods in high-order finite element applications, yet their adoption is traditionally hindered by the prohibitive cost of solving local patch problems. This paper presents a high-performance, matrix-free implementation of a p-multigrid local solver that dismantles the trade-off between smoothing effectiveness and computational efficiency. We focus on the practical realization of this iterative approach, leveraging sum-factorization and explicit SIMD vectorization to minimize memory footprint and maximize arithmetic throughput. The performance analysis demonstrates that the solver effectively hides data-fetching latencies and maintains optimal $\mathcal{O}(p^d)$ memory scaling, even when dominated by geometric data on distorted meshes. The result is a robust smoother that rivals the execution speed of simple pointwise smoothers while preserving the convergence benefits of patch-based methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_02577 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multigrid p-Robustness at Jacobi Speeds: Efficient Matrix-Free Implementation of Local p-Multigrid Solvers Wichrowski, Michał Numerical Analysis Vertex-patch smoothers are essential for the robust convergence of geometric multigrid methods in high-order finite element applications, yet their adoption is traditionally hindered by the prohibitive cost of solving local patch problems. This paper presents a high-performance, matrix-free implementation of a p-multigrid local solver that dismantles the trade-off between smoothing effectiveness and computational efficiency. We focus on the practical realization of this iterative approach, leveraging sum-factorization and explicit SIMD vectorization to minimize memory footprint and maximize arithmetic throughput. The performance analysis demonstrates that the solver effectively hides data-fetching latencies and maintains optimal $\mathcal{O}(p^d)$ memory scaling, even when dominated by geometric data on distorted meshes. The result is a robust smoother that rivals the execution speed of simple pointwise smoothers while preserving the convergence benefits of patch-based methods. |
| title | Multigrid p-Robustness at Jacobi Speeds: Efficient Matrix-Free Implementation of Local p-Multigrid Solvers |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2512.02577 |