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Inf-sup theory for the quasi-static Biot's equations in poroelasticity

C. Kreuzer, P. Zanotti

TL;DR

This paper develops an inf-sup based analysis for the quasi-static Biot's poroelasticity equations in bounded domains. By reformulating the system as an equivalent four-field problem with total pressure $p_{tot}$ and total fluid content $m$, the authors establish existence, uniqueness, and two-sided stability using Banach-Nec̆as-Babuška, with norms that couple the solution components and are robust to the time horizon and material parameters. They derive additional regularity results in space, showing that higher regularity of data can yield higher regularity of the solution under suitable assumptions, and they also analyze the limits of stability through positive and negative results about trial function spaces. A strong-space (spatial) formulation is then introduced to demonstrate a shift of regularity in space, confirming well-posedness and equivalence with the weak formulation under enhanced regularity. Overall, the framework provides a solid foundation for stable discretizations with rigorous a priori and a posteriori error analysis and offers clear paths for extending to more general data and boundary conditions in poroelasticity.

Abstract

We analyze the two-field formulation of the quasi-static Biot's equations in bounded domains by means of the inf-sup theory. For this purpose, we exploit an equivalent four-field formulation of the equations, introducing the so-called total pressure and total fluid content as independent variables. We establish existence, uniqueness and stability of the solution. Our stability estimate is two-sided and robust, meaning that the regularity established for the solution matches the regularity requirements for the data and the involved constants are independent of all material parameters. We prove also that additional regularity in space of the data implies, in some cases, corresponding additional regularity in space of the solution. These results are instrumental to the design and the analysis of discretizations enjoying accurate stability and error estimates.

Inf-sup theory for the quasi-static Biot's equations in poroelasticity

TL;DR

This paper develops an inf-sup based analysis for the quasi-static Biot's poroelasticity equations in bounded domains. By reformulating the system as an equivalent four-field problem with total pressure and total fluid content , the authors establish existence, uniqueness, and two-sided stability using Banach-Nec̆as-Babuška, with norms that couple the solution components and are robust to the time horizon and material parameters. They derive additional regularity results in space, showing that higher regularity of data can yield higher regularity of the solution under suitable assumptions, and they also analyze the limits of stability through positive and negative results about trial function spaces. A strong-space (spatial) formulation is then introduced to demonstrate a shift of regularity in space, confirming well-posedness and equivalence with the weak formulation under enhanced regularity. Overall, the framework provides a solid foundation for stable discretizations with rigorous a priori and a posteriori error analysis and offers clear paths for extending to more general data and boundary conditions in poroelasticity.

Abstract

We analyze the two-field formulation of the quasi-static Biot's equations in bounded domains by means of the inf-sup theory. For this purpose, we exploit an equivalent four-field formulation of the equations, introducing the so-called total pressure and total fluid content as independent variables. We establish existence, uniqueness and stability of the solution. Our stability estimate is two-sided and robust, meaning that the regularity established for the solution matches the regularity requirements for the data and the involved constants are independent of all material parameters. We prove also that additional regularity in space of the data implies, in some cases, corresponding additional regularity in space of the solution. These results are instrumental to the design and the analysis of discretizations enjoying accurate stability and error estimates.
Paper Structure (17 sections, 10 theorems, 146 equations, 3 figures)

This paper contains 17 sections, 10 theorems, 146 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Biot's equations and abstract weak formulation
  3. Two-field formulation
  4. Four-field formulation
  5. Weak formulation
  6. Well-posedness of the weak formulation
  7. Inf-sup stability
  8. Nondegeneracy
  9. Further stability estimates
  10. Positive results
  11. Negative results
  12. Shift of the regularity in space
  13. Strong formulation in space
  14. Well-posedness and regularity
  15. Inf-sup stability
  16. ...and 2 more sections

Key Result

Lemma 3.1

The bilinear form $b$ in E:BiotProblem-abstract-form is such that for all $\widetilde{y}_1 \in \overline{\mathbb{Y}}_1$. The hidden constant depends only on the domain $\Omega$.

Figures (3)

  • Figure 1: Spaces and operators describing the regularity in space for the weak formulation \ref{['E:BiotProblem-weak-formulation']} of the Biot's equations. The symbol '$\equiv$' denotes the identification via the $L^2(\Omega)$-scalar product and $i$ is the embedding operator.
  • Figure 2: Spaces and operators describing the regularity in space for the strong formulation \ref{['E:BiotProblem-strong-formulation']} of the Biot's equations. Note that $i$ denotes the embedding operator.
  • Figure :

Theorems & Definitions (41)

  • Remark 2.1: Parameters
  • Remark 2.2: Initial condition
  • Remark 2.3: Boundary conditions
  • Remark 2.4: Unbounded domains
  • Remark 2.5: Auxiliary variables
  • Remark 2.6: Pressure space
  • Remark 2.7: Definition of $\gamma$
  • Remark 2.8: Two-field formulation
  • Remark 2.9: Norms
  • Remark 2.10: Regularity of $\ell_u$
  • ...and 31 more