Monolithic Algebraic Multigrid Preconditioners for the Stokes Equations
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| author | Voronin, Alexey MacLachlan, Scott Olson, Luke N. Tuminaro, Raymond |
| author_facet | Voronin, Alexey MacLachlan, Scott Olson, Luke N. Tuminaro, Raymond |
| contents | We investigate a novel monolithic algebraic multigrid (AMG) preconditioner for the Taylor-Hood ($\pmb{\mathbb{P}}_2/\mathbb{P}_1$) and Scott-Vogelius ($\pmb{\mathbb{P}}_2/\mathbb{P}_1^{disc}$) discretizations of the Stokes equations. The algorithm is based on the use of the lower-order $\pmb{\mathbb{P}}_1\text{iso}\kern1pt\pmb{\mathbb{P}}_2/\mathbb{P}_1$ operator within a defect-correction setting, in combination with AMG construction of interpolation operators for velocities and pressures. The preconditioning framework is primarily algebraic, though the $\pmb{\mathbb{P}}_1\text{iso}\kern1pt\pmb{\mathbb{P}}_2/\mathbb{P}_1$ operator must be provided. We investigate two relaxation strategies in this setting. Specifically, a novel block factorization approach is devised for Vanka patch systems, which significantly reduces storage requirements and computational overhead, and a Chebyshev adaptation of the LSC-DGS relaxation is developed to improve parallelism. The preconditioner demonstrates robust performance across a variety of 2D and 3D Stokes problems, often matching or exceeding the effectiveness of an inexact block-triangular (or Uzawa) preconditioner, especially in challenging scenarios such as elongated-domain problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_06795 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Monolithic Algebraic Multigrid Preconditioners for the Stokes Equations Voronin, Alexey MacLachlan, Scott Olson, Luke N. Tuminaro, Raymond Numerical Analysis We investigate a novel monolithic algebraic multigrid (AMG) preconditioner for the Taylor-Hood ($\pmb{\mathbb{P}}_2/\mathbb{P}_1$) and Scott-Vogelius ($\pmb{\mathbb{P}}_2/\mathbb{P}_1^{disc}$) discretizations of the Stokes equations. The algorithm is based on the use of the lower-order $\pmb{\mathbb{P}}_1\text{iso}\kern1pt\pmb{\mathbb{P}}_2/\mathbb{P}_1$ operator within a defect-correction setting, in combination with AMG construction of interpolation operators for velocities and pressures. The preconditioning framework is primarily algebraic, though the $\pmb{\mathbb{P}}_1\text{iso}\kern1pt\pmb{\mathbb{P}}_2/\mathbb{P}_1$ operator must be provided. We investigate two relaxation strategies in this setting. Specifically, a novel block factorization approach is devised for Vanka patch systems, which significantly reduces storage requirements and computational overhead, and a Chebyshev adaptation of the LSC-DGS relaxation is developed to improve parallelism. The preconditioner demonstrates robust performance across a variety of 2D and 3D Stokes problems, often matching or exceeding the effectiveness of an inexact block-triangular (or Uzawa) preconditioner, especially in challenging scenarios such as elongated-domain problems. |
| title | Monolithic Algebraic Multigrid Preconditioners for the Stokes Equations |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2306.06795 |