A-posteriori-steered $p$-robust multigrid and domain decomposition methods with optimal step-sizes for mixed finite element discretizations of elliptic problems
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914299921825792 |
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| author | Miraçi, Ani Papež, Jan Vohralík, Martin Yotov, Ivan |
| author_facet | Miraçi, Ani Papež, Jan Vohralík, Martin Yotov, Ivan |
| contents | In this work, we develop algebraic solvers for linear systems arising from the discretization of second-order elliptic partial differential equations by saddle-point mixed finite element methods of arbitrary polynomial degree $p \ge 0$ on possibly highly graded simplicial meshes. We present a multigrid and a two-level domain decomposition approach in two and three space dimensions, steered by a posteriori estimators of the algebraic error. First, we extend [Miraçi, Papež, and Vohralík, SIAM J. Sci. Comput. 43 (2021), S117-S145] to the mixed finite element setting. Extending the multigrid procedure itself is rather natural. To obtain analogous theoretical results, however, a $p$-robust multilevel stable decomposition of the velocity space is needed. In two space dimensions, we can treat the velocity space as the curl of a stream-function Lagrange space, for which the previous results apply. In three space dimensions, we design a novel stable decomposition by combining a one-level high-order local stable decomposition of [Falk and Winther, Found. Comput. Math. (2025), DOI 10.1007/s10208- 025-09700-2] and a multilevel lowest-order stable decomposition of [Hiptmair, Wu, and Zheng, Numer. Math. Theory Methods Appl. 5 (2012), 297-332]. This allows us to prove that our multigrid solver contracts the algebraic error at each iteration $p$-robustly and, simultaneously, that the associated a posteriori estimator is $p$-robustly efficient. Next, we use this multilevel methodology to define a two-level domain decomposition method where the subdomains consist of overlapping patches of coarse-level elements sharing a common coarse-level vertex. We again establish a $p$-robust contraction of the solver and $p$-robust efficiency of the a posteriori estimator. Numerical results presented both for the multigrid approach and the domain decomposition method confirm the theoretical findings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_09872 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A-posteriori-steered $p$-robust multigrid and domain decomposition methods with optimal step-sizes for mixed finite element discretizations of elliptic problems Miraçi, Ani Papež, Jan Vohralík, Martin Yotov, Ivan Numerical Analysis In this work, we develop algebraic solvers for linear systems arising from the discretization of second-order elliptic partial differential equations by saddle-point mixed finite element methods of arbitrary polynomial degree $p \ge 0$ on possibly highly graded simplicial meshes. We present a multigrid and a two-level domain decomposition approach in two and three space dimensions, steered by a posteriori estimators of the algebraic error. First, we extend [Miraçi, Papež, and Vohralík, SIAM J. Sci. Comput. 43 (2021), S117-S145] to the mixed finite element setting. Extending the multigrid procedure itself is rather natural. To obtain analogous theoretical results, however, a $p$-robust multilevel stable decomposition of the velocity space is needed. In two space dimensions, we can treat the velocity space as the curl of a stream-function Lagrange space, for which the previous results apply. In three space dimensions, we design a novel stable decomposition by combining a one-level high-order local stable decomposition of [Falk and Winther, Found. Comput. Math. (2025), DOI 10.1007/s10208- 025-09700-2] and a multilevel lowest-order stable decomposition of [Hiptmair, Wu, and Zheng, Numer. Math. Theory Methods Appl. 5 (2012), 297-332]. This allows us to prove that our multigrid solver contracts the algebraic error at each iteration $p$-robustly and, simultaneously, that the associated a posteriori estimator is $p$-robustly efficient. Next, we use this multilevel methodology to define a two-level domain decomposition method where the subdomains consist of overlapping patches of coarse-level elements sharing a common coarse-level vertex. We again establish a $p$-robust contraction of the solver and $p$-robust efficiency of the a posteriori estimator. Numerical results presented both for the multigrid approach and the domain decomposition method confirm the theoretical findings. |
| title | A-posteriori-steered $p$-robust multigrid and domain decomposition methods with optimal step-sizes for mixed finite element discretizations of elliptic problems |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2406.09872 |