An Enhanced Finite Element Algorithm for Thermal Darcy Flows with Variable Viscosity

This paper deals with the development of a stable and efficient unified finite element method for the numerical solution of thermal Darcy flows with variable viscosity. The governing equations consist of coupling the Darcy equations for the pressure and velocity fields to a convection-diffusion equation for the heat transfer. The viscosity in the Darcy flows is assumed to be nonlinear depending on the temperature of the medium. The proposed method is based on combining a semi-Lagrangian scheme with a Galerkin finite element discretization of the governing equations along with an robust iterative solver for the associate linear systems. The main features of the enhanced finite element algorithm are that the same finite element space is used for all solutions to the problem including the pressure, velocity and temperature. In addition, the convection terms are accurately dealt with using the semi-Lagrangian scheme and the standard Courant-Friedrichs-Lewy condition is relaxed and the time truncation errors are reduced in the diffusion terms. Numerical results are presented for two examples to demonstrate the performance of the proposed finite element algorithm.