Convergence and parameter-robust analysis of several locking-free algorithms for a linear poroelasticity model

Peizhen Wang; Wenlong He; Yajuan Di; Peizhen Wang; Wenlong He; Yajuan Di

doi:10.3934/math.2025527

AIMS Mathematics

2025, Volume 10, Issue 5: 11627-11655. doi: 10.3934/math.2025527

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Research article

Convergence and parameter-robust analysis of several locking-free algorithms for a linear poroelasticity model

1.
School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450046, China
2.
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Received: 26 February 2025 Revised: 26 April 2025 Accepted: 07 May 2025 Published: 21 May 2025
MSC : 65N30, 65M12

The quasi-static linear poroelasticity model is widely applied in science, geophysics, biomechanics, and engineering. The simulations of this model exhibits locking phenomena and may depend on some model parameters. In this work, we follow the idea proposed in [Feng, Ge, and Li, IMA J. Numer. Anal., 2018,330–359] to transform the linear poroelasticity model into a four-field problem. The new four-field problem has a built-in mechanism to circumvent the locking phenomena existing in the original problem. We first prove that the inf-sup condition holds uniformly independent of the model parameters for the four-field problem and design several coupled, decoupled, and BDF2 algorithms in time. After that, we establish the analysis of the unconditionally optimal convergence and parameter robustness for the proposed algorithms. Finally, numerical examples are provided to investigate the convergence and parameter robustness of the proposed algorithms, which have no locking phenomena and are consistent with our theory. In addition, we also apply the algorithms to simulate brain edema, which is aligned with the experiment results.
- poroelasticity model,
- parameter-robust algorithms,
- optimal convergence rate,
- locking phenomena,
- BDF2,
- brain edema
Citation: Peizhen Wang, Wenlong He, Yajuan Di. Convergence and parameter-robust analysis of several locking-free algorithms for a linear poroelasticity model[J]. AIMS Mathematics, 2025, 10(5): 11627-11655. doi: 10.3934/math.2025527

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Abstract

The quasi-static linear poroelasticity model is widely applied in science, geophysics, biomechanics, and engineering. The simulations of this model exhibits locking phenomena and may depend on some model parameters. In this work, we follow the idea proposed in [Feng, Ge, and Li, IMA J. Numer. Anal., 2018,330–359] to transform the linear poroelasticity model into a four-field problem. The new four-field problem has a built-in mechanism to circumvent the locking phenomena existing in the original problem. We first prove that the inf-sup condition holds uniformly independent of the model parameters for the four-field problem and design several coupled, decoupled, and BDF2 algorithms in time. After that, we establish the analysis of the unconditionally optimal convergence and parameter robustness for the proposed algorithms. Finally, numerical examples are provided to investigate the convergence and parameter robustness of the proposed algorithms, which have no locking phenomena and are consistent with our theory. In addition, we also apply the algorithms to simulate brain edema, which is aligned with the experiment results.

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