Block BDDC/FETI-DP Preconditioners for Three-Field mixed finite element Discretizations of Biot's consolidation model
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| Format: | Preprint |
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2025
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| _version_ | 1866915231738888192 |
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| author | Chu, Hanyu Pavarino, Luca Franco Zampini, Stefano |
| author_facet | Chu, Hanyu Pavarino, Luca Franco Zampini, Stefano |
| contents | In this paper, we construct and analyze a block dual-primal preconditioner for Biot's consolidation model approximated by three-field mixed finite elements based on a displacement, pressure, and total pressure formulation. The domain is decomposed into nonoverlapping subdomains, and the continuity of the displacement component across the subdomain interface is enforced by introducing a Lagrange multiplier. After eliminating all displacement variables and the independent subdomain interior components of pressure and total pressure, the problem is reduced to a symmetric positive definite linear system for the subdomain interface pressure, total pressure, and the Lagrange multiplier. This reduced system is solved by a preconditioned conjugate gradient method, with a block dual-primal preconditioner using a Balancing Domain Decomposition by Constraints (BDDC) preconditioner for both the interface total pressure block and the interface pressure blocks, as well as a Finite Element Tearing and Interconnecting-Dual Primal (FETI-DP) preconditioner for the Lagrange multiplier block. By analyzing the conditioning of the preconditioned subsystem associated with the interface pressure and total pressure components, we obtain a condition number bound of the preconditioned system, which is scalable in the number of subdomains, poly-logarithmic in the ratio of subdomain and mesh sizes, and robust with respect to the parameters of the model. Extensive numerical experiments confirm the theoretical result of the proposed algorithm. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_04859 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Block BDDC/FETI-DP Preconditioners for Three-Field mixed finite element Discretizations of Biot's consolidation model Chu, Hanyu Pavarino, Luca Franco Zampini, Stefano Numerical Analysis In this paper, we construct and analyze a block dual-primal preconditioner for Biot's consolidation model approximated by three-field mixed finite elements based on a displacement, pressure, and total pressure formulation. The domain is decomposed into nonoverlapping subdomains, and the continuity of the displacement component across the subdomain interface is enforced by introducing a Lagrange multiplier. After eliminating all displacement variables and the independent subdomain interior components of pressure and total pressure, the problem is reduced to a symmetric positive definite linear system for the subdomain interface pressure, total pressure, and the Lagrange multiplier. This reduced system is solved by a preconditioned conjugate gradient method, with a block dual-primal preconditioner using a Balancing Domain Decomposition by Constraints (BDDC) preconditioner for both the interface total pressure block and the interface pressure blocks, as well as a Finite Element Tearing and Interconnecting-Dual Primal (FETI-DP) preconditioner for the Lagrange multiplier block. By analyzing the conditioning of the preconditioned subsystem associated with the interface pressure and total pressure components, we obtain a condition number bound of the preconditioned system, which is scalable in the number of subdomains, poly-logarithmic in the ratio of subdomain and mesh sizes, and robust with respect to the parameters of the model. Extensive numerical experiments confirm the theoretical result of the proposed algorithm. |
| title | Block BDDC/FETI-DP Preconditioners for Three-Field mixed finite element Discretizations of Biot's consolidation model |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2504.04859 |