GMRES Convergence Analysis for Nonsymmetric Saddle-Point Systems When the Field of Values Contains the Origin
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915793314250752 |
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| author | Chen, Hao Greif, Chen |
| author_facet | Chen, Hao Greif, Chen |
| contents | We present a field-of-values (FOV) analysis for preconditioned nonsymmetric saddle-point linear systems, where zero is included in the field of values of the matrix. We rely on recent results of Crouzeix and Greenbaum [Spectral sets: numerical range and beyond. SIAM Journal on Matrix Analysis and Applications, 40(3):1087-1101, 2019], showing that a convex region with a circular hole is a spectral set. Sufficient conditions are derived for convergence independent of the matrix dimensions. We apply our results to preconditioned nonsymmetric saddle-point systems, and show their applicability to families of block preconditioners that have not been previously covered by existing FOV analysis. A limitation of our theory is that the preconditioned matrix is required to have a small skew-symmetric part in norm. Consequently, our analysis may not be applicable, for example, to fluid flow problems characterized by a small viscosity coefficient. Some numerical results illustrate our findings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07156 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | GMRES Convergence Analysis for Nonsymmetric Saddle-Point Systems When the Field of Values Contains the Origin Chen, Hao Greif, Chen Numerical Analysis We present a field-of-values (FOV) analysis for preconditioned nonsymmetric saddle-point linear systems, where zero is included in the field of values of the matrix. We rely on recent results of Crouzeix and Greenbaum [Spectral sets: numerical range and beyond. SIAM Journal on Matrix Analysis and Applications, 40(3):1087-1101, 2019], showing that a convex region with a circular hole is a spectral set. Sufficient conditions are derived for convergence independent of the matrix dimensions. We apply our results to preconditioned nonsymmetric saddle-point systems, and show their applicability to families of block preconditioners that have not been previously covered by existing FOV analysis. A limitation of our theory is that the preconditioned matrix is required to have a small skew-symmetric part in norm. Consequently, our analysis may not be applicable, for example, to fluid flow problems characterized by a small viscosity coefficient. Some numerical results illustrate our findings. |
| title | GMRES Convergence Analysis for Nonsymmetric Saddle-Point Systems When the Field of Values Contains the Origin |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2505.07156 |