# A Stabilized Hybrid Mixed Finite Element Method for Poroelasticity

## TL;DR

This paper introduces a stabilized hybrid mixed finite element method for Biot's poroelasticity model, improving stability and efficiency through face bubble functions, elimination techniques, and robust preconditioners, with confirmed numerical results.

## Contribution

It develops a stabilized hybrid scheme with face bubble functions for Biot's model, ensuring stability, optimal error estimates, and efficient preconditioning, addressing previous stability issues.

## Key findings

- Stable hybrid scheme with face bubble functions
- Elimination of bubble functions and Darcy velocity
- Numerical experiments confirm theoretical stability and efficiency

## Abstract

In this work, we consider a hybrid mixed finite element method for Biot's model. The hybrid P1-RT0-P0 discretization of the displacement-pressure-Darcy's velocity system of Biot's model presented in \cite{C. Niu} is not uniformly stable with respect to the physical parameters, resulting in some issues in numerical simulations. To alleviate such problems, following \cite{V. Girault}, we stabilize the hybrid scheme with face bubble functions and show the well-posedness with respect to physical and discretization parameters, which provide optimal error estimates of the stabilized method. We introduce a perturbation of the bilinear form of the displacement which allows for the elimination of the bubble functions. Together with eliminating Darcy's velocity by hybridization, we obtain an eliminated system whose size is the same as the classical P1-RT0-P0 discretization. Based on the well-posedness of the eliminated system, we design block preconditioners that are parameter-robust. Numerical experiments are presented to confirm the theoretical results of the stabilized scheme as well as the block preconditioners.