A locking-free numerical method for the quasi-static linear thermo-poroelasticity model

Abstract

In this paper, we propose a locking-free numerical method to solve the quasi-static linear thermo-poroelasticity model, which exhibits two types of locking phenomena: Poisson locking and nonphysical oscillations. By analyzing the regularity of solution of the original model, we find that the Poisson locking is caused by divu≈0 as λ→+∞. Moreover, when discretizing the diffusion equations using backward Euler method for time, one can see that divu1≈0 for some special parameters. If we directly use the continuous Galerkin mixed finite element method (FEM) to solve the original model, the pressure and temperature fields exhibit numerical oscillations at early times. To overcome these two locking phenomena, we introduce a new variable ξ=αp+βT−λdivu to reformulate the original problem into a new one. This new problem includes a built-in mechanism to maintain stability for the continuous Galerkin mixed FEM. We further prove the existence and uniqueness of the weak solution by using the standard Galerkin method in conjunction with regularity estimates. We also design a fully discrete time-stepping scheme that employs mixed FEM with P2−P1−P1−P1 element pairs for the space variables and the backward Euler method for the time variable. Optimal convergence is demonstrated in both space and time. Finally, numerical examples are provided to demonstrate the optimal convergence rates of the variables and the robustness of the proposed method with respect to ν, and to verify the absence of the locking phenomenon.