(untitled) (Journal of Computational Mathematics, 2026)
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.