Multi-Level Mixed Finite Element Methods Based on Different Iterations for the Steady Boussinesq Problem

Abstract

Three multi-level mixed finite element methods for the steady Boussinesq equations are analyzed and discussed in this paper. The nonlinear and multi-variables coupled problem on a coarse mesh with the mesh size $h_0$ is solved firstly, and then, a series of decoupled and linear subproblems with the Stokes, Oseen and Newton iterations are solved on the successive and refined grids with the mesh sizes $h_j, j = 1, 2, . . . , J$. The computational scales are reduced and the computational costs are saved. Furthermore, the uniform stability and convergence results in both $L^2$- and $H^1$-norms of are derived under some uniqueness conditions by using the mathematical induction and constructing the dual problems. Theoretical results show that the multi-level methods have the same order of numerical solutions in the $H^1$-norm as the one level method with the mesh sizes $h_j = h^2_{j−1}$, $j = 1, 2, . . . , J$. Finally, some numerical results are provided to investigate and compare the effectiveness of the multi-level mixed finite element methods.