Monolithic two-level Schwarz preconditioner for Biot's consolidation model in two space dimensions
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| Format: | Preprint |
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2024
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| _version_ | 1866910423306993664 |
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| author | Meggendorfer, Stefan Kanschat, Guido Kraus, Johannes |
| author_facet | Meggendorfer, Stefan Kanschat, Guido Kraus, Johannes |
| contents | This paper addresses the construction and analysis of a class of domain decomposition methods for the iterative solution of the quasi-static Biot problem in three-field formulation. The considered discrete model arises from time discretization by the implicit Euler method and space discretization by a family of strongly mass-conserving methods exploiting $H^{div}$-conforming approximations of the solid displacement and fluid flux fields. For the resulting saddle-point problem, we construct monolithic overlapping domain decomposition (DD) methods whose analysis relies on a transformation into an equivalent symmetric positive definite system and on stable decompositions of the involved finite element spaces under proper problem-dependent norms. Numerical results on two-dimensional test problems are in accordance with the provided theoretical uniform convergence estimates for the two-level multiplicative Schwarz method. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_16684 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Monolithic two-level Schwarz preconditioner for Biot's consolidation model in two space dimensions Meggendorfer, Stefan Kanschat, Guido Kraus, Johannes Numerical Analysis 65M12, 65M55, 65F10 This paper addresses the construction and analysis of a class of domain decomposition methods for the iterative solution of the quasi-static Biot problem in three-field formulation. The considered discrete model arises from time discretization by the implicit Euler method and space discretization by a family of strongly mass-conserving methods exploiting $H^{div}$-conforming approximations of the solid displacement and fluid flux fields. For the resulting saddle-point problem, we construct monolithic overlapping domain decomposition (DD) methods whose analysis relies on a transformation into an equivalent symmetric positive definite system and on stable decompositions of the involved finite element spaces under proper problem-dependent norms. Numerical results on two-dimensional test problems are in accordance with the provided theoretical uniform convergence estimates for the two-level multiplicative Schwarz method. |
| title | Monolithic two-level Schwarz preconditioner for Biot's consolidation model in two space dimensions |
| topic | Numerical Analysis 65M12, 65M55, 65F10 |
| url | https://arxiv.org/abs/2404.16684 |