Mixed and multipoint finite element methods for rotation-based poroelasticity

TL;DR

This paper introduces a novel multipoint mixed finite element method for Biot poroelasticity that employs auxiliary variables and hybridization, ensuring stability, convergence, and parameter-robust preconditioning, validated through numerical experiments.

Contribution

It develops a new four-field formulation and a multipoint rotation-flux mixed finite element method for Biot equations, enhancing stability and efficiency.

Findings

The methods are stable and convergent in weighted norms.

Numerical experiments confirm theoretical stability and convergence.

The approach provides parameter-robust preconditioners.

Abstract

This work proposes a mixed finite element method for the Biot poroelasticity equations that employs the lowest-order Raviart-Thomas finite element space for the solid displacement and piecewise constants for the fluid pressure. The method is based on the formulation of linearized elasticity as a weighted vector Laplace problem. By introducing the solid rotation and fluid flux as auxiliary variables, we form a four-field formulation of the Biot system, which is discretized using conforming mixed finite element spaces. The auxiliary variables are subsequently removed from the system in a local hybridization technique to obtain a multipoint rotation-flux mixed finite element method. Stability and convergence of the four-field and multipoint mixed finite element methods are shown in terms of weighted norms, which additionally leads to parameter-robust preconditioners. Numerical experiments confirm the theoretical results.