Analysis of multiphysics finite element method for quasi-static thermo-poroelasticity with a nonlinear convective transport term

TL;DR

This paper develops a stable multiphysics finite element method for quasi-static thermo-poroelasticity with nonlinear convective transport, using reformulation, iterative algorithms, and convergence analysis to improve numerical stability and accuracy.

Contribution

It introduces a novel reformulation and iterative algorithm for stable, convergent finite element solutions of nonlinear thermo-poroelastic models with thermal convection.

Findings

The proposed method effectively overcomes numerical oscillations.

The method achieves optimal convergence order.

Convergence is proven using Banach fixed point theorem.

Abstract

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.