A coupling of Galerkin and mixed finite element methods for the quasi-static thermo-poroelasticity with nonlinear convective transport

Abstract

In this paper, we propose a combined Galerkin and mixed finite element methods to analyze the fully coupled nonlinear thermo-poroelastic model problems. We design the Galerkin finite element method for the temperature, the mixed finite element method for the pressure, the Galerkin finite element method for the elastic displacement. We linearize the nonlinear convective transport term in the energy balance equation and establish the fully discrete finite element schemes. The stability and convergence of the coupled method are obtained. In particular, all previous works have required certain time step restrictions, but we unconditionally prove the optimal error estimates without certain extra restrictions on both time step and spatial meshes. Finally, some numerical examples are presented to illustrate the accuracy of the method and confirm the unconditional stability of the method.