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Inf-sup stable discretization of the quasi-static Biot's equations in poroelasticity

C. Kreuzer, P. Zanotti

TL;DR

This work presents an inf-sup stable full discretization for the quasi-static Biot's equations in poroelasticity using a four-field formulation with total pressure $p_ ext{tot}$ and total fluid content $m$. A backward-Euler time discretization coupled with a space-discrete variational framework preserves the problem's stability structure, and a Lagrange FE space is shown to yield quasi-optimal error with constants robust to material parameters. The authors also develop time and space interpolation operators that enable decoupling of the best-approximation error into standard spatial and temporal components, and prove convergence with first-order rates under mild regularity assumptions. Numerical tests on Terzaghi's problem and a cantilever-bracket scenario confirm the theory, illustrating stable behavior and meaningful error decay under mesh refinement.

Abstract

We propose a new full discretization of the Biot's equations in poroelasticity. The construction is driven by the inf-sup theory, which we recently developed. It builds upon the four-field formulation of the equations obtained by introducing the total pressure and the total fluid content. We discretize in space with Lagrange finite elements and in time with backward Euler. We establish inf-sup stability and quasi-optimality of the proposed discretization, with robust constants with respect to all material parameters. We further construct an interpolant showing how the error decays for smooth solutions.

Inf-sup stable discretization of the quasi-static Biot's equations in poroelasticity

TL;DR

This work presents an inf-sup stable full discretization for the quasi-static Biot's equations in poroelasticity using a four-field formulation with total pressure and total fluid content . A backward-Euler time discretization coupled with a space-discrete variational framework preserves the problem's stability structure, and a Lagrange FE space is shown to yield quasi-optimal error with constants robust to material parameters. The authors also develop time and space interpolation operators that enable decoupling of the best-approximation error into standard spatial and temporal components, and prove convergence with first-order rates under mild regularity assumptions. Numerical tests on Terzaghi's problem and a cantilever-bracket scenario confirm the theory, illustrating stable behavior and meaningful error decay under mesh refinement.

Abstract

We propose a new full discretization of the Biot's equations in poroelasticity. The construction is driven by the inf-sup theory, which we recently developed. It builds upon the four-field formulation of the equations obtained by introducing the total pressure and the total fluid content. We discretize in space with Lagrange finite elements and in time with backward Euler. We establish inf-sup stability and quasi-optimality of the proposed discretization, with robust constants with respect to all material parameters. We further construct an interpolant showing how the error decays for smooth solutions.
Paper Structure (21 sections, 16 theorems, 180 equations, 9 figures, 1 table, 1 algorithm)

This paper contains 21 sections, 16 theorems, 180 equations, 9 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Inf-sup theory for the Biot's equations
  3. Initial-boundary value problem
  4. Weak Formulation and well-posedness
  5. Abstract inf-sup stable discretization
  6. Abstract discretisation in space
  7. Discretization in time
  8. Full discretization
  9. Well-posedness
  10. Inf-sup stability
  11. Discretisation with Lagrange elements in space
  12. Concrete space discretisation
  13. Quasi-optimality
  14. A priori error estimates
  15. Time interpolation
  16. ...and 6 more sections

Key Result

Theorem 2.4

For all possible data $(\ell_u, \ell_p, \ell_0) \in L^2(\mathbb{U}^*) \times L^2({\mathbb{P}}^*) \times {\mathbb{P}}^*$, the weak formulation E:BiotProblem-weak-formulation has a unique solution $y_1 = (u,p_\text{tot}, p, m) \in \overline{\mathbb{Y}}_1$, which satisfies the two-sided stability bound Moreover, we have $m \in C^0({\mathbb{P}}^*)$ as well as the norm equivalence All hidden constants

Figures (9)

  • Figure 2.1: Spaces and operators describing the regularity in space for the weak formulation \ref{['E:BiotProblem-weak-formulation']} of the Biot's equations. The triple lines on the bottom indicate identification via the $L^2(\Omega)$-scalar product.
  • Figure 3.1: Spaces and operators describing the regularity in space for the discretization of the Biot's equations. The triple lines on the bottom indicate identification via the $L^2(\Omega)$-scalar product.
  • Figure 4.1: Commutative diagram representing the relation between the time interpolants. The second component in the operator $(\,\mathcal{J} ,\;(\cdot)(0) \,)$ on the left is the evaluation at $t=0$.
  • Figure 4.2: Commutative diagrams representing the relations among the space interpolants.
  • Figure 5.1: Boundary conditions (left) and computational mesh (right) for the cantilever bracket problem.
  • ...and 4 more figures

Theorems & Definitions (54)

  • Remark 2.1: Notation
  • Remark 2.2: Functional setting
  • Remark 2.3: Auxiliary variables
  • Theorem 2.4: Well-posedness of the weak formulation
  • proof
  • Remark 2.5: Trial functions
  • Remark 2.6: Functions and functionals
  • Remark 3.1: Notation for the discretization
  • Remark 3.2: Discretization of the Hilbert triplet
  • Remark 3.3: Spurious pressure oscillations
  • ...and 44 more