Multiphysics Finite Element Method for Quasi-Static Thermo-Poroelasticity

摘要

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 (P2-P1documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$P_2-P_1$$end{document} element pair) for the variables of udocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${textbf {u}}$$end{document} and ξdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$xi $$end{document}, P1documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$P_1$$end{document}-conforming element for the variable of ηdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$eta $$end{document} and P1documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$P_1$$end{document}-conforming element for the variable of γdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$gamma $$end{document}, 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.