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Optimal error estimates of sequential finite element method for nonlinear thermo-poroelasticity problems
Fang Wang, Fan Chen, Changqing Lv, Chenguang Zhou
TL;DR
The paper addresses efficient numerical solution of nonlinear, fully coupled quasi-static thermo-poroelasticity systems with convective transport. It develops a three-step sequential finite element method with backward Euler time discretization and analyzes its well-posedness and stability. The method avoids internal iterations, achieves optimal spatial and temporal convergence estimates, and is supported by numerical experiments, subject to a conditional time-step stability restriction for the sequential scheme.
Problem
Fully implicit methods for coupled thermo-poroelasticity require large-scale systems, motivating an efficient approach for nonlinear problems with convective transport.
Method
A three-step sequential decoupling algorithm solves pressure, temperature, and displacement successively using finite elements and backward Euler time discretization.
Results
The analysis establishes well-posedness, stability, and optimal convergence rates, while numerical tests show optimal spatial orders and robustness across coupling strengths and extreme conductivities.
Takeaways & Limitations
The method solves once per time step without internal iterations while retaining the reported optimal convergence behavior.
Takeaways & Limitations
The sequential scheme has a conditional stability requirement τ ≤ 2θma0 M 2, unlike the coupled scheme's unconditional stability.
Abstract
from arXiv · showhide
This study introduces and analyzes a three-step sequential decoupling algorithm designed to address nonlinear, fully coupled quasi-static thermo-poroelasticity systems incorporating convective transport. The finite element method is employed for spatial discretization and the backward Euler method for temporal discretization. The proposed sequential method has a higher computing efficiency than the fully implicit nonlinear numerical scheme, since it does not require any internal iterations. The well-posedness of the numerical solution is discussed by introducing a cut-off operator and the stability analysis of the algorithm is performed. Rigorous analysis yields optimal convergence order estimates for both spatial and temporal discretizations. In order to confirm the theoretical results and the effectiveness of the suggested approach, numerical experiments are finally carried out.
1 Introduction
The paper targets nonlinear, fully coupled quasi-static thermo-poroelasticity with convective transport, where fully implicit methods are computationally expensive. It proposes a three-step sequential finite element scheme with backward Euler time discretization, stability analysis, optimal error estimates, and numerical validation.
- The model couples mechanical equilibrium, fluid pressure, and temperature through stress, storage, thermal, and convective transport terms.
- Fully implicit thermo-poroelasticity methods couple all variables at each time step, requiring large-scale algebraic systems and incurring high computational expense.
- The proposed method sequentially solves for pressure, temperature, and displacement using values from previous time layers and previously computed fields.The thermo-poroelasticity model is separated at every time step.
- Finite elements discretize space and backward Euler discretizes time, while the analysis addresses stability, existence, uniqueness, and optimal convergence rates.
- The sequential method solves once at each time step without internal iterations, unlike fully implicit or iterative alternatives.
2 Coupled algorithm
The coupled finite element scheme is formulated under structural and regularity assumptions, analyzed through energy methods, and discretized on regular meshes with backward Euler time stepping. Its fully discrete version is proved unconditionally energy-stable under a stated parameter condition.
- Assumptions: The model assumes time-independent symmetric positive-definite permeability and thermal-conductivity tensors, positive material parameters, compatibility inequalities, regular data, and sufficient solution smoothness.
- Variational formulation: The variational formulation couples displacement, pressure, and temperature through mechanical, storage, diffusion, and convective bilinear terms.
- Energy analysis: The continuous weak solution satisfies an energy dissipation law combining the energy derivative, pressure diffusion, temperature diffusion, and source terms.
- Discretization: The finite element construction uses a regular triangulation, polynomial discrete spaces, projection operators, and backward Euler time stepping.
- Analysis tools: The cut-off operator is introduced to facilitate convergence analysis and agrees with the exact Darcy flux when that flux is bounded and the cut-off level is sufficiently large.
- Stability: The fully discrete coupled scheme is unconditionally energy-stable when θm = 2MCp.
3 Sequentially decoupled algorithm
The proposed sequential algorithm decouples the thermo-poroelastic system into linear subproblems while establishing well-posedness, stability, and optimal spatial-temporal error estimates under stated assumptions.
- Algorithm and analysis: The algorithm solves the coupled problem through sequential subproblems, with existence, uniqueness, stability, and optimal convergence analyzed for the resulting numerical solution.The analysis uses a stabilizer to address coupling in the fully decoupled formulation.
- Stabilization and stability: The stabilization parameter γ is independent of mesh and time-step sizes and depends only on material parameters.The theoretical choice is selected to ensure positivity in the stability derivation, while numerical tests suggest the practical bound can be relaxed.
- Algorithm and analysis: Initialization requires computing the first pressure approximation separately because the sequential formulations depend on prior temperature and displacement values.A linear numerical scheme is used to compute the initial pressure approximation and obtain its optimal convergence order.
- Stabilization and stability: The sequential scheme is conditionally stable under a time-step restriction, unlike the unconditional stability established for the coupled scheme.The restriction is identified as sharper than the coupled formulation’s unconditional stability condition.
- Error estimates: Under the stated regularity and parameter assumptions, the analysis derives optimal convergence-order estimates in both spatial and temporal discretizations.The proof combines error equations, projection estimates, Taylor expansion, and discrete energy arguments.
4 Numerical simulations
Numerical experiments test temporal and spatial convergence, computational efficiency, stabilization sensitivity, and robustness across coupling and physical-parameter regimes. The sequential method achieves the predicted optimal convergence behavior, matches fully implicit FEM accuracy, and is computationally more efficient.
- Experimental setup: The experiments use exact-solution tests on a uniform triangular mesh across five parameter configurations with varying coupling strengths.The configurations PA1–PA5 are designed to test different coupling intensities.
- Convergence order: First-order temporal convergence in the L2 norm is observed for displacement, pressure, and temperature under PA1.The spatial mesh is fixed at h = 1/128 for this test.
- Convergence order: Under τ = h2, the sequential method achieves optimal spatial convergence orders across all five coupling strengths.The results validate the theoretical predictions of Theorem 3.3.
- Computational efficiency: SFEM and fully implicit FEM have comparable error accuracy and convergence orders at the same grid scale.The comparison is reported through Figure 1.
- Computational efficiency: SFEM is computationally more efficient than fully implicit FEM under PA1.The runtime comparison is reported in Table 8.
- Stabilization sensitivity: For α = β = 1, displacement converges at O(h) in H1 and pressure and temperature at O(h2) in L2 across tested γ values.These rates remain nearly identical across γ values spanning three orders of magnitude below the theoretical value.
- Parameter robustness: Optimal convergence rates persist under stronger coupling, extreme K and Θ values, and a0 = b0 = c0 = 0.The displacement error grows slightly with stronger coupling, while the reported convergence rates remain optimal.