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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References
Bercovier, J., Pironneau, O.: Error estimates for finite element solution of the Stokes problem in the primitive variables. Numer. Math. 211–244, 33 (1979)
Brun, M.K., Berre, I., Nordbotten, J.M., Radu, F.A.: Upscaling of the coupling of hydromechanical and thermal processes in quasi-static poroelastic medium. Transp. Porous Media 137–158, 124 (2018)
Boal, N., Gaspar, F., Vabishchevich, P.: Finite-difference analysis for the linear thermoporoelasticity problem and its numerical resolution by multigrid methods. Math. Model. Anal. 227–244, 17 (2012)
Butler, R.M.: Thermal Recovery of Oil and Bitumen. Englewood Cliffs, N.J., Prentice Hall (1991)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problem. North-Holland, Amsterdam (1978)
Coussy, O.: Poromechancis. John Wiley & sons, England (2004)
Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 1. Springer, New York (1990)
Feng, X.B., Ge, Z.H. , Li, Y.K.: Analysis of a multiphysics finite element method for a poroelasticity model. IMA J. Numer. Anal., 330-359, 38(2018). arXiv:1411.7464, [math.NA], 2014
Feng, X.B., He, Y.N.: Fully discrete finite element approximations of a polymer gel model. SIAM J. Numer. Anal. 2186–2217, 48 (2010)
Gatmiri, B., Delage, P.: A formulation of fully coupled thermal-hydraulic-mechanical behaviour of saturated porous media-numerical approach. Int. J. Numer. Anal. Meth. Geomech. 199–225, 21 (1997)
Hamley, I.: Introduction to Soft Matter. John Wiley, New York (2007)
Kaviany, M.: Principles of Heat Transfer in Porous Media. Springer, Berlin (1991)
Kolesov, A.E., Vabishchevich, P.N., Vasilyeva, M.V.: Splitting schemes for poroelasticity and thermoelasticity problems. Comput. Math. Appl. 2185–2198, 67 (2014)
Lee, C.K., Mei, C.C.: Thermal consolidation in porous media by homogenization theory-I. Derivation of macroscale equations. Adv. Water Resour. 20, 127–144 (1997)
Mu, M., Zhu, X.H.: Decoupled schemes for a non-stationary mixed stokes-darcy model. Math. Comput. 707–731, 79 (2009)
Phillips, P.J., Wheeler, M.F.: A coupling of mixed and continuous Galerkin finite element methods for poroelasticity II: the discrete in time case. Comput. Geosci. 145–158, 11 (2007)
Phillips, P.J., Wheeler, M.F.: Overcoming the problem of locking in linear elasticity and poroelasticity: an heuristic approach. Comput. Geosci. 1–15, 13 (2009)
Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer-Verlag, New York (1997)
Roberts, J.E., Thomas, J.M.: Mixed and hybrid methods. Hand. Numer. Anal. 523–639, 2 (1991)
Schowalter, R.E.: Diffusion in poro-elastic media. J. Math. Anal. 310–340, 251 (2000)
Suvorov, A.P., Selvadurai, A.P.: Macroscopic constitutive equations of thermo-poroviscoelasticity derived using eigenstrains. J. Mech. Phys. Solids 1461–1473, 58 (2010)
Tanaka, T., Fillmore, D.J.: Kinetics of swelling of gels. J. Chem. Phys. 1214–1218, 70 (1979)
Temam, R.: Navier-Stokes Equations: Theory and Numerical Analysis. American Mathematical Society, AMS Chelsea Publishing, Providence, RI (2000)
Terzaghi, K.: Theoretical Soil Mechanics. John Wiley & Sons Inc, New York (1943)
Vabishchevich, P.N., Vasil’eva, M.V., Kolesov, A.E.: Splitting scheme for poroelasticity and thermoelasticity problems. Comput. Math. Math. Phys. 1305–1315, 54 (2014)
Yi, S.Y.: A study of two modes of locking in poroelasticity. SIAM J. Numer. Anal. 1915–1936, 55 (2017)
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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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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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DOI: https://doi.org/10.1007/s10915-022-01877-w