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

Abstract This paper considers the two-level iterative finite element methods for the steady natural 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...
Ausführliche Beschreibung

Abstract This paper considers the two-level iterative finite element methods for the steady natural 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 %$h\ll H%$ 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 %$h={\mathcal {O}}(H^2)%$ 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 condition. Finally, some numerical experiments are designed to confirm the established theoretical findings and verify the performance of the proposed numerical schemes. Ausführliche Beschreibung