In this paper, we propose a multiphysics finite element method for a quasi-static thermo-poroelasticity model with a nonlinear convective transport term. To design some stable numerical methods and reveal the multi-physical processes of deformation, diffusion and heat, we introduce three new variables to reformulate the original model into a fluid coupled problem. Then, we introduce an Newton's iterative algorithm by replacing the convective transport term with $\nabla T^{i}\cdot(\bm{K}\nabla p^{i-1})$, $\nabla T^{i-1}\cdot(\bm{K}\nabla p^{i})$ and $\nabla T^{i-1}\cdot(\bm{K}\nabla p^{i-1})$, and apply the Banach fixed point theorem to prove the convergence of the proposed method. Then, we propose a multiphysics finite element method with Newton's iterative algorithm, which is equivalent to a stabilized method, can effectively overcome the numerical oscillation caused by the nonlinear thermal convection term. Also, we prove that the fully discrete multiphysics finite element method has an optimal convergence order. Finally, we draw conclusions to summarize the main results of this paper.

翻译：本文针对含非线性对流输运项的准静态热孔隙弹性模型，提出了一种多物理场有限元方法。为设计稳定的数值方法并揭示变形、扩散与传热的多物理过程，我们引入三个新变量将原始模型重构为流体耦合问题。随后，通过将对流输运项替换为$\nabla T^{i}\cdot(\bm{K}\nabla p^{i-1})$、$\nabla T^{i-1}\cdot(\bm{K}\nabla p^{i})$及$\nabla T^{i-1}\cdot(\bm{K}\nabla p^{i-1})$，引入牛顿迭代算法，并应用巴拿赫不动点定理证明所提方法的收敛性。继而，我们提出一种等效于稳定化方法的牛顿迭代多物理场有限元方法，能有效克服非线性热对流项引发的数值振荡。同时证明全离散多物理场有限元方法具有最优收敛阶。最后，通过结论总结本文主要成果。