Monolithic two-level Schwarz preconditioner for Biot's consolidation model in two space dimensions
Stefan Meggendorfer, Guido Kanschat, Johannes Kraus
TL;DR
The paper develops a monolithic two-level overlapping Schwarz preconditioner for the quasi-static Biot consolidation model discretized with mass-conserving $H^{\text{div}}$-conforming methods in two space dimensions. By transforming the saddle-point system into an equivalent singularly perturbed SPD problem and establishing stable space decompositions, the authors prove uniform convergence of both additive and multiplicative Schwarz iterations under mild mesh-size relations and bounded permeability. Theoretical results are complemented by 2D numerical experiments demonstrating parameter-robust performance across a range of Lamé, permeability, and storage coefficients, with a coarse-space solve that scales favorably. The work advances robust preconditioning for poroelastic systems and lays groundwork for extensions to 3D and high-frequency/contrast settings.
Abstract
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.