Towards a Multigrid Preconditioner Interpretation of Hierarchical Poincaré-Steklov Solvers
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912985404932096 |
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| author | Lorca, J. P. Lucero |
| author_facet | Lorca, J. P. Lucero |
| contents | We revisit the Hierarchical Poincaré-Steklov (HPS) method in a preconditioned iterative setting for variable-coefficient Helmholtz problems with impedance boundary conditions. HPS is commonly presented as a direct solver based on nested dissection and high-order tensor-product discretizations; here we recast its hierarchical merge tree as a multilevel preconditioner for the assembled skeleton (trace) system. The main goal is to flexibilize the final, memory-intensive coarse stage of direct HPS by replacing the exact coarse solve with a small, fixed amount of iterative work, thereby exposing tunable trade-offs between memory footprint and time to solution. Numerical experiments on a two-dimensional scattering benchmark illustrate these trade-offs and compare against both unpreconditioned GMRES and the classic direct HPS pipeline with an exact coarse space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_00735 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Towards a Multigrid Preconditioner Interpretation of Hierarchical Poincaré-Steklov Solvers Lorca, J. P. Lucero Numerical Analysis 65N22, 65N35, 65N55, 65F05 We revisit the Hierarchical Poincaré-Steklov (HPS) method in a preconditioned iterative setting for variable-coefficient Helmholtz problems with impedance boundary conditions. HPS is commonly presented as a direct solver based on nested dissection and high-order tensor-product discretizations; here we recast its hierarchical merge tree as a multilevel preconditioner for the assembled skeleton (trace) system. The main goal is to flexibilize the final, memory-intensive coarse stage of direct HPS by replacing the exact coarse solve with a small, fixed amount of iterative work, thereby exposing tunable trade-offs between memory footprint and time to solution. Numerical experiments on a two-dimensional scattering benchmark illustrate these trade-offs and compare against both unpreconditioned GMRES and the classic direct HPS pipeline with an exact coarse space. |
| title | Towards a Multigrid Preconditioner Interpretation of Hierarchical Poincaré-Steklov Solvers |
| topic | Numerical Analysis 65N22, 65N35, 65N55, 65F05 |
| url | https://arxiv.org/abs/2511.00735 |