Newton iterative method based on finite element discretization for the stationary Darcy–Brinkman equations

Abstract

The stationary Darcy–Brinkman equations in the double-diffusive convection, which model the heat and mass transfer phenomena, are considered in this paper. Based on a suitable contractive operator, the existence and uniqueness of the problem are firstly proved by using the fixed point theorem. The regularities of the weak solution are also derived. Then, the Newton iterative method is studied for solving the nonlinear discrete system generated from the finite element approximation, including the stability and the optimal error estimates regarding the spatial mesh size and the iterative factor. The analysis indicates that the viscosity coefficient has more impact on the numerical algorithm than the thermal conductivity and the mass diffusivity coefficients. Finally, many numerical examples are shown to confirm the correctness of the theoretical prediction.