Splitting-strategies for arbitrary-order fully mixed finite element discretizations of the Biot equations

TL;DR

This paper develops and analyzes fully mixed finite element methods for the Biot equations, enabling flexible, stable discretizations that preserve conservation laws and demonstrate convergence, with effective splitting strategies and negative stabilization.

Contribution

It introduces a novel fully mixed formulation with weak symmetry enforcement, establishes stability and error estimates, and proves the effectiveness of negative stabilization in splitting strategies.

Findings

Stable mixed finite element discretizations for Biot equations are achieved.

Optimal a priori error estimates are established for arbitrary order spaces.

Numerical results confirm the theoretical convergence and effectiveness of the proposed methods.

Abstract

We study the fully mixed formulation of the Biot equations, which is characterized by a symmetric coupling between flow and deformation. This structure enables the use of stable mixed finite elements for each subproblem without a strong compatibility condition across the two subphysics. To exploit this flexibility while preserving the conservation structure of both subproblems, we consider fully mixed finite element methods in which the symmetry of the elastic stress tensor is enforced weakly. The resulting mixed formulation exhibits a saddle-point structure whose stability is determined by suitable inf--sup conditions. Inf--sup stability is established for several families of discrete spaces of arbitrary order, leading to optimal a priori error estimates. Iterative splitting strategies following the classical fixed-stress split with additional tuning are specifically investigated for the fully mixed formulation, with proof of convergence and rates depending on the coupling strength. Contrary to previous analyses on coupled problems with a symmetric structure, we theoretically prove the efficacy of negative stabilization, consistent with Schur-complement ideas. Numerical results based on analytical solutions and the classical Mandel problem support the theory.