Parameter-robust preconditioners for a stabilized control-volume discretization of Biot's equations

Abstract

We develop and analyze parameter-robust block preconditioners for a stabilized control-volume discretization of the quasi-static Biot equations. Beginning with a five-field formulation, we eliminate the poroelastic stress and time-scaled Darcy velocity through independent vertex-local solves, obtaining a symmetric cell-centered displacement--rotation--pressure system. A macrocell rotation-jump term stabilizes the weak-symmetry constraint. We prove the well-posedness of the fully discrete scheme and show that the condensed, coupled displacement--rotation mechanics block is uniformly spectrally equivalent to its block-diagonal counterpart. For uncoupled preconditioners, we derive an exact discrete pressure-control criterion and a sufficient parameter condition that accommodates the scaling \(c_0\asymp K_{\rm dr}^{-1}\), thereby allowing the storage coefficient to vanish in the nearly incompressible limit. We then construct a Schur-consistent pressure block that approximates the mechanics-induced pressure correction and prove its uniform equivalence to the exact pressure Schur complement. These results yield parameter-independent spectral bounds for block-diagonal preconditioning and weighted field-of-values estimates for lower-triangular preconditioning. Practical realizations employ a macrocell mass-matrix approximation for the stabilized rotation block and tailored two-grid solvers for the displacement and pressure blocks. Numerical experiments verify the expected convergence behavior in two dimensions and demonstrate robust performance under mesh refinement and broad variations of the physical parameters in both two and three dimensions.