Monolithic two-level Schwarz preconditioner for Biot's consolidation model in two space dimensions
<|reference_start|>Monolithic two-level Schwarz preconditioner for Biot's consolidation model in two space dimensions: 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.<|reference_end|>
@article{meggendorfer2024monolithic,
    title={Monolithic two-level Schwarz preconditioner for Biot's consolidation
  model in two space dimensions},
    author={Stefan Meggendorfer, Guido Kanschat, Johannes Kraus},
    journal={arXiv preprint arXiv:2404.16684},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.16684},
    primaryClass={math.NA cs.NA}
}