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Some semi-decoupled algorithms with optimal convergence for a four-field linear thermo-poroelastic model

Ziliang Li, Mingchao Cai, Jingzhi Li, Qiang Liu

TL;DR

The paper develops three non-iterative semi-decoupled solvers for a linear four-field thermo-poroelastic system, comprising two sequential schemes and one parallel scheme. Each method splits the problem into a mixed elasticity subproblem and a coupled pressure–temperature subproblem, while maintaining unconditional stability and optimal convergence; the initial step remains fully coupled. A rigorous a priori error analysis substantiates the convergence claims, and extensive numerical tests demonstrate robustness across wide parameter regimes, including nearly incompressible and nearly degenerate cases. The parallel algorithm, in particular, achieves true concurrency with substantial computational savings, making these methods well-suited for large-scale geomechanics, biomedical, and energy applications.

Abstract

We propose three semi-decoupled algorithms for efficiently solving a four-field thermoporoelastic model. The first two algorithms adopt a sequential strategy: at the initial time step, all variables are computed simultaneously using a monolithic solver; thereafter, the system is split into a mixed linear elasticity subproblem and a coupled pressure-temperature reaction-diffusion subproblem. The two variants differ in the order in which these subproblems are solved. To further improve computational efficiency, we introduce a parallel semidecoupled algorithm. In this approach, the four-field system is solved monolithically only at the first time step, and the two subproblems are then solved in parallel at subsequent time levels. None of the three algorithms requires iterative procedures at each time step, and are free from stabilization. Rigorous analysis confirms their unconditional stability, optimal convergence rates, and robustness under a wide range of physical parameter settings. These theoretical results are further validated by numerical experiments.

Some semi-decoupled algorithms with optimal convergence for a four-field linear thermo-poroelastic model

TL;DR

The paper develops three non-iterative semi-decoupled solvers for a linear four-field thermo-poroelastic system, comprising two sequential schemes and one parallel scheme. Each method splits the problem into a mixed elasticity subproblem and a coupled pressure–temperature subproblem, while maintaining unconditional stability and optimal convergence; the initial step remains fully coupled. A rigorous a priori error analysis substantiates the convergence claims, and extensive numerical tests demonstrate robustness across wide parameter regimes, including nearly incompressible and nearly degenerate cases. The parallel algorithm, in particular, achieves true concurrency with substantial computational savings, making these methods well-suited for large-scale geomechanics, biomedical, and energy applications.

Abstract

We propose three semi-decoupled algorithms for efficiently solving a four-field thermoporoelastic model. The first two algorithms adopt a sequential strategy: at the initial time step, all variables are computed simultaneously using a monolithic solver; thereafter, the system is split into a mixed linear elasticity subproblem and a coupled pressure-temperature reaction-diffusion subproblem. The two variants differ in the order in which these subproblems are solved. To further improve computational efficiency, we introduce a parallel semidecoupled algorithm. In this approach, the four-field system is solved monolithically only at the first time step, and the two subproblems are then solved in parallel at subsequent time levels. None of the three algorithms requires iterative procedures at each time step, and are free from stabilization. Rigorous analysis confirms their unconditional stability, optimal convergence rates, and robustness under a wide range of physical parameter settings. These theoretical results are further validated by numerical experiments.

Paper Structure

This paper contains 12 sections, 7 theorems, 118 equations, 2 figures, 7 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Mathematical model and its reformulation
  3. A linear thermo-poroelasticity model
  4. Numerical algorithms
  5. The coupled algorithm
  6. Two sequential semi-decoupled algorithms
  7. A parallel semi-decoupled algorithm
  8. Convergence analysis
  9. A priori estimate for the parallel semi-decoupled algorithm
  10. A priori estimate for the split sequential algorithms
  11. Numerical experiments
  12. Conclusions

Key Result

Lemma 4.1

Let $B$ be a symmetric bilinear form. Then is satisfied, which immediately implies the following inequality

Figures (2)

  • Figure 1: Injection-production domain
  • Figure 2: Example 2: displacement $\bm u$(left), pressure $p$(middle), and temperature $T$ at terminal time $\tau=1$

Theorems & Definitions (12)

  • Remark 1
  • Remark 2
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • proof
  • Theorem 4.1: estimate for Algorithm 3
  • proof
  • Remark 3
  • ...and 2 more