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

Fleurianne Bertrand, Jakub Wiktor Both, Tugay Dağlı

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.

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

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.
Paper Structure (14 sections, 4 theorems, 108 equations, 3 figures, 4 tables)

This paper contains 14 sections, 4 theorems, 108 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Stress-based porous media equations
  3. Derivation of the mixed formulation
  4. Finite element formulation
  5. Finite elements on simplicial meshes
  6. Finite elements on tensor-product meshes
  7. Iterative coupling for the fully-discrete system
  8. Fixed-stress split
  9. Step 1: Flow
  10. Step 2: Mechanics
  11. Numerical results
  12. Manufactured solution on the unit square
  13. Mandel's problem
  14. Concluding remarks

Key Result

Theorem 1

Let $\Omega$ be a bounded, connected Lipschitz domain. Assume that the Lamé parameters $\mu,\lambda$, the storage coefficient $\tilde{c}_0$, and the Biot parameter $\tilde{\alpha}$ are strictly positive, and that the permeability tensor $\boldsymbol\kappa$ is symmetric and uniformly elliptic. Given holds.

Figures (3)

  • Figure 1: Spatial convergence for the manufactured solution with $\gamma_1=\gamma_2=1$ and $\beta=0$. The errors are evaluated at the final time $t=1$ for discrete space orders $k=1,2,3$.
  • Figure 2: Schematic illustration of Mandel's problem and the reduced computational quarter domain used in the simulations.
  • Figure 3: Dimensionless solution variables for Mandel's problem.

Theorems & Definitions (12)

  • Theorem 1
  • Lemma 1
  • proof
  • Remark 1
  • Lemma 2
  • proof
  • Remark 2: Standard fixed-stress split
  • Lemma 3: Convergence of a tuned fixed-stress split
  • proof
  • Remark 3: Convergence of all fields
  • ...and 2 more