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Multiphysics Finite Element Method for Quasi-Static Thermo-Poroelasticity

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Abstract

In this paper, we propose a multiphysics finite element method for the quasi-static thermo-poroelasticity model with small Péclet number. To reveal the multi-physical processes of deformation, diffusion and heat transfer, we reformulate the original model into a fluid coupled problem. Then, we prove the existence and uniqueness of a weak solution to the original problem and the reformulated problem. And we propose a fully discrete finite element method based on the multiphysics reformulation–Taylor-Hood element (\(P_2-P_1\) element pair) for the variables of \({\textbf {u}}\) and \(\xi \), \(P_1\)-conforming element for the variable of \(\eta \) and \(P_1\)-conforming element for the variable of \(\gamma \), and the backward Euler method for time discretization. And we give the stability analysis of the above proposed method, also we prove that the fully discrete multiphysics finite element method has an optimal convergence order. Finally, we show some numerical examples to verify the theoretical results.

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Funding

The work was supported by the National Natural Science Foundation of China under grant No. 11971150, the Major Projects of International Science and Technology Cooperation of Henan University under grant No. 2021ybxm07 and the Cultivation Project of First Class Subject of Henan University under grant No. 2019YLZDJL08.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University, Kaifeng, 475004, People’s Republic of China

    Yuxiang Chen

  2. School of Mathematics and Statistics, Henan Engineering Research Center for Artificial Intelligence Theory and Algorithms, Henan University, Kaifeng, 475004, People’s Republic of China

    Zhihao Ge

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  1. Yuxiang Chen
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Correspondence to Zhihao Ge.

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The work of this author was supported by the National Natural Science Foundation of China under grant No. 11971150, the Major Projects of International Science and Technology Cooperation of Henan University under grant No. 2021ybxm07 and the Cultivation Project of First Class Subject of Henan University under grant No. 2019YLZDJL08.

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Chen, Y., Ge, Z. Multiphysics Finite Element Method for Quasi-Static Thermo-Poroelasticity. J Sci Comput 92, 43 (2022). https://doi.org/10.1007/s10915-022-01877-w

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