Monolithic Multigrid Preconditioners for High-Order Discretizations of Stokes Equations
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909249499561984 |
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| author | Voronin, Alexey Harper, Graham MacLachlan, Scott Olson, Luke N. Tuminaro, Raymond S. |
| author_facet | Voronin, Alexey Harper, Graham MacLachlan, Scott Olson, Luke N. Tuminaro, Raymond S. |
| contents | This work introduces and assesses the efficiency of a monolithic $ph$MG multigrid framework designed for high-order discretizations of stationary Stokes systems using Taylor-Hood and Scott-Vogelius elements. The proposed approach integrates coarsening in both approximation order ($p$) and mesh resolution ($h$), to address the computational and memory efficiency challenges that are often encountered in conventional high-order numerical simulations. Our numerical results reveal that $ph$MG offers significant improvements over traditional spatial-coarsening-only multigrid ($h$MG) techniques for problems discretized with Taylor-Hood elements across a variety of problem sizes and discretization orders. In particular, the $ph$MG method exhibits superior performance in reducing setup and solve times, particularly when dealing with higher discretization orders and unstructured problem domains. For Scott-Vogelius discretizations, while monolithic $ph$MG delivers low iteration counts and competitive solve phase timings, it exhibits a discernibly slower setup phase when compared to a multilevel (non-monolithic) full-block-factorization (FBF) preconditioner where $ph$MG is employed only for the velocity unknowns. This is primarily due to the setup costs of the larger mixed-field relaxation patches with monolithic $ph$MG versus the patch setup costs with a single unknown type for FBF. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07253 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Monolithic Multigrid Preconditioners for High-Order Discretizations of Stokes Equations Voronin, Alexey Harper, Graham MacLachlan, Scott Olson, Luke N. Tuminaro, Raymond S. Numerical Analysis This work introduces and assesses the efficiency of a monolithic $ph$MG multigrid framework designed for high-order discretizations of stationary Stokes systems using Taylor-Hood and Scott-Vogelius elements. The proposed approach integrates coarsening in both approximation order ($p$) and mesh resolution ($h$), to address the computational and memory efficiency challenges that are often encountered in conventional high-order numerical simulations. Our numerical results reveal that $ph$MG offers significant improvements over traditional spatial-coarsening-only multigrid ($h$MG) techniques for problems discretized with Taylor-Hood elements across a variety of problem sizes and discretization orders. In particular, the $ph$MG method exhibits superior performance in reducing setup and solve times, particularly when dealing with higher discretization orders and unstructured problem domains. For Scott-Vogelius discretizations, while monolithic $ph$MG delivers low iteration counts and competitive solve phase timings, it exhibits a discernibly slower setup phase when compared to a multilevel (non-monolithic) full-block-factorization (FBF) preconditioner where $ph$MG is employed only for the velocity unknowns. This is primarily due to the setup costs of the larger mixed-field relaxation patches with monolithic $ph$MG versus the patch setup costs with a single unknown type for FBF. |
| title | Monolithic Multigrid Preconditioners for High-Order Discretizations of Stokes Equations |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2407.07253 |