An efficient two-step algorithm for steady-state natural convection problem

In this paper, we propose a new highly efficient two-step algorithm based on local Gauss integration for the 2D steady-state natural convection problem. The basic idea of the algorithm is to compute an initial approximation for the velocity, pressure and temperature based on a lowest equal-order finite element pair P1-P1-P1, then to solve a linear system based on a quadratic equal-order finite element pair P2-P2-P2 on the same mesh. Next, we give the corresponding stability and convergence of the algorithm, which show that the new two-step algorithm has the same order convergence rate as the quadratic equal-order stabilized finite element method. Finally, some numerical examples show that the new method is efficient, reliable, has good precision and can save a lot of computational time compared with the quadratic equal-order stabilized method.