A fast spectral overlapping domain decomposition method with discretization-independent conditioning bounds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917050093404160 |
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| author | Dirckx, Simon Yesypenko, Anna Martinsson, Per-Gunnar |
| author_facet | Dirckx, Simon Yesypenko, Anna Martinsson, Per-Gunnar |
| contents | A domain decomposition method for the solution of general variable-coefficient elliptic partial differential equations on regular domains is introduced. The method is based on tessellating the domain into overlapping thin slabs or shells, and then explicitly forming a reduced linear system that connects the different domains. Rank-structure ('H-matrix structure') is exploited to handle the large dense blocks that arise in the reduced linear system. Importantly, the formulation used is well-conditioned, as it converges to a second kind Fredholm equation as the precision in the local solves is refined. Moreover, the dense blocks that arise are far more data-sparse than in existing formulations, leading to faster and more efficient H-matrix arithmetic. To form the reduced linear system, black-box randomized compression is used, taking full advantage of the fact that sparse direct solvers are highly efficient on the thin sub-domains. Numerical experiments demonstrate that our solver can handle oscillatory 2D and 3D problems with as many as 28 million degrees of freedom. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_25991 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A fast spectral overlapping domain decomposition method with discretization-independent conditioning bounds Dirckx, Simon Yesypenko, Anna Martinsson, Per-Gunnar Numerical Analysis Computational Engineering, Finance, and Science Mathematical Physics A domain decomposition method for the solution of general variable-coefficient elliptic partial differential equations on regular domains is introduced. The method is based on tessellating the domain into overlapping thin slabs or shells, and then explicitly forming a reduced linear system that connects the different domains. Rank-structure ('H-matrix structure') is exploited to handle the large dense blocks that arise in the reduced linear system. Importantly, the formulation used is well-conditioned, as it converges to a second kind Fredholm equation as the precision in the local solves is refined. Moreover, the dense blocks that arise are far more data-sparse than in existing formulations, leading to faster and more efficient H-matrix arithmetic. To form the reduced linear system, black-box randomized compression is used, taking full advantage of the fact that sparse direct solvers are highly efficient on the thin sub-domains. Numerical experiments demonstrate that our solver can handle oscillatory 2D and 3D problems with as many as 28 million degrees of freedom. |
| title | A fast spectral overlapping domain decomposition method with discretization-independent conditioning bounds |
| topic | Numerical Analysis Computational Engineering, Finance, and Science Mathematical Physics |
| url | https://arxiv.org/abs/2510.25991 |