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

Three multi-level mixed finite element methods for steady Boussinesq equations are analyzed and discussed in this paper. We first solve the nonlinear multi-variable coupled problem on coarse meshes with mesh size $h_0$, and then solve a sequence of decoupled linear subproblems via Stokes, Oseen and Newton iterations on hierarchically refined meshes with mesh sizes $h_j$, $j=1,2,\dots,J$. The proposed strategies effectively reduce computational scale and lower computational expenses. Furthermore, uniform stability and convergence results in both $L^2$-norm and $H^1$-norm are derived under certain uniqueness conditions by virtue of mathematical induction and dual problem techniques. Theoretical analysis indicates that when the mesh satisfies $h_j=h_{j-1}^2$ for $j=1,2,\dots,J$, the multi-level methods possess the same $H^1$-norm convergence order as the standard single-level method. Finally, numerical examples are presented to investigate and compare the performance of these multi-level mixed finite element approaches.