Unified analysis of saddle point problems via auxiliary space theory
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917380677959680 |
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| author | Park, Jongho |
| author_facet | Park, Jongho |
| contents | We present sharp estimates for the extremal eigenvalues of the Schur complements arising in saddle point problems. These estimates are derived using the auxiliary space theory, in which a given iterative method is interpreted as an equivalent but more elementary iterative method on an auxiliary space, enabling us to obtain sharp convergence estimates. The proposed framework improves or refines several existing results, which can be recovered as corollaries of our results. To demonstrate the versatility of the framework, we present various applications from scientific computing: the augmented Lagrangian method, mixed finite element methods, and nonoverlapping domain decomposition methods. In all these applications, the condition numbers of the corresponding Schur complements can be estimated in a straightforward manner using the proposed framework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_11434 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Unified analysis of saddle point problems via auxiliary space theory Park, Jongho Numerical Analysis 65F10, 65N22, 65N55 We present sharp estimates for the extremal eigenvalues of the Schur complements arising in saddle point problems. These estimates are derived using the auxiliary space theory, in which a given iterative method is interpreted as an equivalent but more elementary iterative method on an auxiliary space, enabling us to obtain sharp convergence estimates. The proposed framework improves or refines several existing results, which can be recovered as corollaries of our results. To demonstrate the versatility of the framework, we present various applications from scientific computing: the augmented Lagrangian method, mixed finite element methods, and nonoverlapping domain decomposition methods. In all these applications, the condition numbers of the corresponding Schur complements can be estimated in a straightforward manner using the proposed framework. |
| title | Unified analysis of saddle point problems via auxiliary space theory |
| topic | Numerical Analysis 65F10, 65N22, 65N55 |
| url | https://arxiv.org/abs/2509.11434 |