The paper describes a new algebraically defined multilevel preconditioner for the finite element matrix arising from the discretization of a second-order selfadjoint elliptic boundary value problem. The construction of the preconditioner is based on a sequence of grids (levels) given by successive refinement and the corresponding hierarchical two-by-two partitioning on the nodal basis stiffness matrices on each level. The preconditioner is described recursively and exploits an approximate factorization of the partitioned matrices into block triangular factors with approximating the Schur complement by the stiffness matrix or a preconditioner on the precending (coaser) level. Moreover, after each \(k_ 0\) levels the preconditioner is corrected by a polynomial approximation. Due to the polynomial correction, the relative condition number of the preconditioner is shown to be uniformly bounded with respect to the number of levels and the preconditioner is shown to be of optimal order of complexity for both 2D and 3D problem domains. In addition, an adaptive implementation of the preconditioner is proposed and numerical experiments are performed.