Uniformly well-posed hybridized discontinuous Galerkin/hybrid mixed discretizations for Biot's consolidation model

• Published in: arXiv.org 2020-12
• Main Authors: Kraus, Johannes, Lederer, Philip L, Lymbery, Maria, Schöberl, Joachim
• Format: Article
• Language: English
• Subjects: Computational fluid dynamics; Conservation; Consolidation; Galerkin method; Iterative methods; Parameter estimation; Parameter robustness; Robustness (mathematics); Seepage; Well posed problems
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2012.08584

We consider the quasi-static Biot's consolidation model in a three-field formulation with the three unknown physical quantities of interest being the displacement \(\boldsymbol{u}\) of the solid matrix, the seepage velocity \(\boldsymbol{v}\) of the fluid and the pore pressure \(p\). As conservation of fluid mass is a leading physical principle in poromechanics, we preserve this property using an \(\boldsymbol{H}(\operatorname{div})\)-conforming ansatz for \(\boldsymbol{u}\) and \(\boldsymbol{v}\) together with an appropriate pressure space. This results in Stokes and Darcy stability and exact, that is, pointwise mass conservation of the discrete model. The proposed discretization technique combines a hybridized discontinuous Galerkin method for the elasticity subproblem with a mixed method for the flow subproblem, also handled by hybridization. The latter allows for a static condensation step to eliminate the seepage velocity from the system while preserving mass conservation. The system to be solved finally only contains degrees of freedom related to \(\boldsymbol{u}\) and \(p\) resulting from the hybridization process and thus provides, especially for higher-order approximations, a very cost-efficient family of physics-oriented space discretizations for poroelasticity problems. We present the construction of the discrete model, theoretical results related to its uniform well-posedness along with optimal error estimates and parameter-robust preconditioners as a key tool for developing uniformly convergent iterative solvers. Finally, the cost-efficiency of the proposed approach is illustrated in a series of numerical tests for three-dimensional test cases.