Two-level iterative finite element methods for the stationary natural convection equations with different viscosities based on three corrections

This paper considers the two-level iterative finite element methods for the steady natu-ral convection equations under some uniqueness conditions with the Simple-, Oseen-and Newton-type corrections. Firstly, the stability and convergence of the one-level iterative finite element methods are analyzed under some restrictions on physical parameters. Secondly, under the strong uniqueness condition, we develop the two-level finite element method with Simple, Oseen and Newton iterations of m times on the coarse mesh tau H with mesh size H, and then, the considered problem is linearized in three correction schemes with the Simple, Oseen and Newton corrections one time on the fine grid tau h with mesh size h << H based on the obtained iterative solutions. From the theoretical point of view, the results obtained by the two-level iterative methods have the same precision as those obtained by the one-level method which mesh sizes satisfy h = O(H-2) and the iterative steps are greater than some constants. Thirdly, the stability and convergence of one-level Oseen iterative scheme with respect to the mesh size and the iterative time m are provided under a weak uniqueness condi-tion. Finally, some numerical experiments are designed to confirm the established theoretical findings and verify the performance of the proposed numerical schemes.