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A Unified Model for Thermo- and Multiple-Network Poroelasticity with a Global-in-Time Iterative Decoupling Scheme

Huipeng Gu, Mingchao Cai, Jingzhi Li, Yu Jiang

Abstract

This paper introduces a unified model for thermo-poroelasticity and multiple-network poroelasticity, reformulated into a total-pressure-based system. We first establish the well-posedness of the problem via a Galerkin-based argument and subsequently introduce a robust space-time finite element approximation. To efficiently solve the fully coupled system, we propose a global-in-time iterative algorithm that sequentially decouples the mechanics from the transport equations, while incorporating necessary stabilization terms. We explicitly analyze the convergence rate and provide a rigorous proof that the proposed scheme constitutes a contraction mapping under physically relevant conditions, thereby ensuring its unconditional convergence. Numerical experiments confirm the theoretical stability bounds and demonstrate optimal convergence rates in both space and time, yielding solutions free of non-physical pressure oscillations.

A Unified Model for Thermo- and Multiple-Network Poroelasticity with a Global-in-Time Iterative Decoupling Scheme

Abstract

This paper introduces a unified model for thermo-poroelasticity and multiple-network poroelasticity, reformulated into a total-pressure-based system. We first establish the well-posedness of the problem via a Galerkin-based argument and subsequently introduce a robust space-time finite element approximation. To efficiently solve the fully coupled system, we propose a global-in-time iterative algorithm that sequentially decouples the mechanics from the transport equations, while incorporating necessary stabilization terms. We explicitly analyze the convergence rate and provide a rigorous proof that the proposed scheme constitutes a contraction mapping under physically relevant conditions, thereby ensuring its unconditional convergence. Numerical experiments confirm the theoretical stability bounds and demonstrate optimal convergence rates in both space and time, yielding solutions free of non-physical pressure oscillations.

Paper Structure

This paper contains 21 sections, 7 theorems, 118 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mathematical model
  3. Model specializations and reformulation
  4. Weak formulation
  5. Space-time iterative approximation
  6. Step 1: Transport sub-problem
  7. Step 2: Mechanics sub-problem
  8. Fully-discrete approximations
  9. Spatial and temporal discretization
  10. Fully-discrete monolithic algorithm
  11. Global-in-time iterative decoupled algorithm
  12. Step 1: Discrete transport sub-problem
  13. Step 2: Discrete mechanics sub-problem
  14. Numerical experiments
  15. Convergence verification
  16. ...and 6 more sections

Key Result

Theorem 1

Under Assumption assumption1, System weakform admits a unique solution

Figures (3)

  • Figure 1: Convergence behavior of the relative iterative errors
  • Figure 2: Cross-section of the generalized pressures $\phi$ and $\psi$ in the generalized Barry-Mercer problem.
  • Figure 3: Numerical solution of the generalized Barry-Mercer problem at $t=\pi/2$.

Theorems & Definitions (13)

  • Theorem 1: Well-posedness
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • proof
  • Lemma 3
  • Theorem 3
  • proof
  • ...and 3 more