In Example 11:
// Reposition the submatrices...  The idea is this:
      //         -           -          -  -
      //        | Kuu Kuv Kup |        | Fu |
      //   Ke = | Kvu Kvv Kvp |;  Fe = | Fv |
      //        | Kpu Kpv Kpp |        | Fv |
      //         -           -          -  -
      // The \p DenseSubMatrix.repostition () member takes the
      // (row_offset, column_offset, row_size, column_size).
      // Similarly, the \p DenseSubVector.reposition () member
      // takes the (row_offset, row_size)
      Kuu.reposition (u_var*n_u_dofs, u_var*n_u_dofs, n_u_dofs, n_u_dofs);
      Kuv.reposition (u_var*n_u_dofs, v_var*n_u_dofs, n_u_dofs, n_v_dofs);
      Kup.reposition (u_var*n_u_dofs, p_var*n_u_dofs, n_u_dofs, n_p_dofs);
      Kvu.reposition (v_var*n_v_dofs, u_var*n_v_dofs, n_v_dofs, n_u_dofs);
      Kvv.reposition (v_var*n_v_dofs, v_var*n_v_dofs, n_v_dofs, n_v_dofs);
      Kvp.reposition (v_var*n_v_dofs, p_var*n_v_dofs, n_v_dofs, n_p_dofs);
      Kpu.reposition (p_var*n_u_dofs, u_var*n_u_dofs, n_p_dofs, n_u_dofs);
      Kpv.reposition (p_var*n_u_dofs, v_var*n_u_dofs, n_p_dofs, n_v_dofs);
      Kpp.reposition (p_var*n_u_dofs, p_var*n_u_dofs, n_p_dofs, n_p_dofs);

      Fu.reposition (u_var*n_u_dofs, n_u_dofs);
      Fv.reposition (v_var*n_u_dofs, n_v_dofs);
      Fp.reposition (p_var*n_u_dofs, n_p_dofs);
I don't understand these code. What are the structures of Total Dof Vector and Element Dof Vector?  For example in 2D, the freedom of node is u , v.  The total Dof Vector is like [u_1,u_2,...u_N, v_1, v_2,...,  v_N], or like [u_1, v_1, u_2, v_2, ..., u_N, v_N], ( N is the node number).

Suggestions: The examples are very usefully to help you understand libMesh. It was more better that the govering equation, the discrete finite element formulation are given.