This paper deals with the iterative solution of large sparse symmetric
positive definite systems. We investigate preconditioning techniques
of the two-level type that are based on a block factorization of the s
ystem matrix. Whereas the basic scheme assumes an exact inversion of t
he submatrix related to the first block of unknowns, we analyze the ef
fect of using an approximate inverse instead. We derive condition numb
er estimates that are valid for any type of approximation of the Schur
complement and that do not assume the use of the hierarchical basis.
They show that the two-level methods are stable when using approximate
inverses based on modified ILU techniques, or explicit inverses that
meet some raw-sum criterion. On the other hand, we bring to the light
that the use of standard approximate inverses based on convergent spli
ttings can have a dramatic effect on the convergence rate. These concl
usions are numerically illustrated on some examples.