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

Johannes Kraus, Philip L. Lederer*, Maria Lymbery, Joachim Schöberl

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

7 Citations (Scopus)

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 u of the solid matrix, the seepage velocity 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 H(div)-conforming ansatz for u and 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 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.

Original languageEnglish
Article number113991
JournalComputer methods in applied mechanics and engineering
Volume384
DOIs
Publication statusPublished - 1 Oct 2021
Externally publishedYes

Keywords

  • Biot's consolidation model
  • Hybrid discontinuous Galerkin methods
  • Hybrid mixed methods
  • Norm-equivalent preconditioners
  • Parameter-robust LBB stability
  • Strongly mass conserving high-order discretizations
  • n/a OA procedure

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