Towards Matrix-Free Patch Smoothers for the Stokes Problem: Evaluating Local p-Multigrid Solvers
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908775125876736 |
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| author | Wichrowski, Michał |
| author_facet | Wichrowski, Michał |
| contents | Vertex-patch smoothers offer an effective strategy for achieving robust geometric multigrid convergence for the Stokes equations, particularly in the context of high-order finite elements. However, their practical efficiency is often limited by the computational cost of solving the local saddle-point problems, especially when explicit matrix factorizations are not feasible. We explore a fully iterative, matrix-free-compatible approach to the local patch solve using $p$-multigrid techniques. We evaluate different local solver configurations: Braess-Sarazin and block-triangular preconditioners. Our numerical experiments suggest that the Braess-Sarazin approach is particularly resilient. We find that a single iteration of the local solver yields global convergence rates comparable to those obtained with exact local solvers, even on distorted meshes and in the presence of large viscosity jumps. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_13230 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Towards Matrix-Free Patch Smoothers for the Stokes Problem: Evaluating Local p-Multigrid Solvers Wichrowski, Michał Numerical Analysis 65Y10, 65Y20, 65N55, 65N30 Vertex-patch smoothers offer an effective strategy for achieving robust geometric multigrid convergence for the Stokes equations, particularly in the context of high-order finite elements. However, their practical efficiency is often limited by the computational cost of solving the local saddle-point problems, especially when explicit matrix factorizations are not feasible. We explore a fully iterative, matrix-free-compatible approach to the local patch solve using $p$-multigrid techniques. We evaluate different local solver configurations: Braess-Sarazin and block-triangular preconditioners. Our numerical experiments suggest that the Braess-Sarazin approach is particularly resilient. We find that a single iteration of the local solver yields global convergence rates comparable to those obtained with exact local solvers, even on distorted meshes and in the presence of large viscosity jumps. |
| title | Towards Matrix-Free Patch Smoothers for the Stokes Problem: Evaluating Local p-Multigrid Solvers |
| topic | Numerical Analysis 65Y10, 65Y20, 65N55, 65N30 |
| url | https://arxiv.org/abs/2601.13230 |