Preconditioning of a hybridizable discontinuous Galerkin method for Biot's consolidation model

TL;DR

This paper develops a parameter-robust preconditioner for a hybridizable discontinuous Galerkin method applied to Biot's consolidation model, ensuring efficiency and stability across various parameters in both 2D and 3D cases.

Contribution

The paper introduces a new parameter-robust preconditioner for HDG discretizations of Biot's model, including a reduced form applicable after static condensation.

Findings

Preconditioner is robust across parameters in 2D and 3D.

Reduced preconditioner maintains robustness post-static condensation.

Numerical experiments confirm effectiveness and stability.

Abstract

We present a parameter-robust preconditioner for a hybridizable discontinuous Galerkin (HDG) discretization of a four-field formulation of Biot's consolidation model. We first determine a parameter-robust preconditioner for the full discretization. HDG methods, however, allow for static condensation. We therefore apply the framework presented in our previous work [arXiv:2503.05918, 2025] to show that a reduced form of the preconditioner is also parameter-robust for the reduced HDG discretization. We verify the parameter-robustness of the preconditioner through numerical examples in both two and three dimensions.