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

TL;DR

This paper introduces a stable, mass-conserving hybridized discontinuous Galerkin/hybrid mixed discretization for Biot's consolidation model, offering a cost-effective and robust approach for simulating poroelasticity with optimal error estimates.

Contribution

It develops a novel hybridized discretization method combining DG and mixed techniques for Biot's model, ensuring uniform well-posedness and mass conservation, with efficient solvers and error analysis.

Findings

The method guarantees pointwise mass conservation.

It achieves optimal error estimates and parameter-robust preconditioning.

Numerical tests confirm cost-efficiency and accuracy in 3D cases.

Abstract

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.