A geometrically informed algebraic multigrid preconditioned iterative approach for solving high-order finite element systems
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914205365436416 |
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| author | Xu, Songzhe Rasouli, Majid Kirby, Robert M. Moxey, David Sundar, Hari |
| author_facet | Xu, Songzhe Rasouli, Majid Kirby, Robert M. Moxey, David Sundar, Hari |
| contents | Algebraic multigrid (AMG) is conventionally applied in a black-box fashion, agnostic to the underlying geometry. In this work, we propose that using geometric information -- when available -- to assist with setting up the AMG hierarchy is beneficial, especially for solving linear systems resulting from high-order finite element discretizations. High-order problems draw considerable interest to both the scientific and engineering communities, but lack efficient solvers, at least open-source codes, tailored for unstructured high-order discretizations targeting large-scale, real-world applications. For geometric multigrid, it is known that using p-coarsening before h-coarsening can provide better scalability, but setting up p-coarsening is non-trivial in AMG. We develop a geometrically informed algebraic multigrid (GIAMG) method, as well as an associated high-performance computing program, which is able to set up a grid hierarchy that includes p-coarsening at the top grids with minimal information of the geometry from the user. A major advantage of using p-coarsening with AMG -- beyond the benefits known in the context of geometric multigrid (GMG) -- is the increased sparsification of coarse grid operators. We extensively evaluate GIAMG by testing on the 3D Helmholtz and incompressible flow problems, and demonstrate mesh-independent convergence, and excellent parallel scalability. We also compare the performance of GIAMG with existing AMG packages, including Hypre and ML. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_15121 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A geometrically informed algebraic multigrid preconditioned iterative approach for solving high-order finite element systems Xu, Songzhe Rasouli, Majid Kirby, Robert M. Moxey, David Sundar, Hari Numerical Analysis Algebraic multigrid (AMG) is conventionally applied in a black-box fashion, agnostic to the underlying geometry. In this work, we propose that using geometric information -- when available -- to assist with setting up the AMG hierarchy is beneficial, especially for solving linear systems resulting from high-order finite element discretizations. High-order problems draw considerable interest to both the scientific and engineering communities, but lack efficient solvers, at least open-source codes, tailored for unstructured high-order discretizations targeting large-scale, real-world applications. For geometric multigrid, it is known that using p-coarsening before h-coarsening can provide better scalability, but setting up p-coarsening is non-trivial in AMG. We develop a geometrically informed algebraic multigrid (GIAMG) method, as well as an associated high-performance computing program, which is able to set up a grid hierarchy that includes p-coarsening at the top grids with minimal information of the geometry from the user. A major advantage of using p-coarsening with AMG -- beyond the benefits known in the context of geometric multigrid (GMG) -- is the increased sparsification of coarse grid operators. We extensively evaluate GIAMG by testing on the 3D Helmholtz and incompressible flow problems, and demonstrate mesh-independent convergence, and excellent parallel scalability. We also compare the performance of GIAMG with existing AMG packages, including Hypre and ML. |
| title | A geometrically informed algebraic multigrid preconditioned iterative approach for solving high-order finite element systems |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2512.15121 |